Datasets:
Unnamed: 0.1 int64 0 899 | Unnamed: 0 int64 0 899 | links stringlengths 31 199 | thms stringlengths 8 5.1k | proofs stringlengths 12 27.3k |
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0 | 0 | https://proofwiki.org//wiki/Primitive_of_Square_of_Arcsine_of_x_over_a | $\ds \int \paren {\arcsin \frac x a}^2 \rd x = x \paren {\arcsin \frac x a}^2 - 2 x + 2 \sqrt{a^2 - x^2} \arcsin \frac x a + C$ | Let:
\(\ds u\) \(=\) \(\ds \arcsin \frac x a\) \(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sin u\) \(=\) \(\ds \frac x a\) Definition of Real Arcsine \(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \cos u\) \(=\) \(\ds \sqrt {1 - \frac {x^2} {a^2} }... |
1 | 1 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosecant_Function | $\ds \int \csc^2 x \rd x = -\cot x + C$
where $C$ is an arbitrary constant . | From Derivative of Cotangent Function :
$\dfrac \d {\d x} \cot x = -\csc^2 x$
The result follows from the definition of primitive .
$\blacksquare$ |
2 | 2 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosecant_of_a_x | $\ds \int \csc^2 a x \rd x = -\frac {\cot a x} a + C$ | \(\ds \int \csc^2 x \rd x\) \(=\) \(\ds -\cot x + C\) Primitive of $\csc^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \csc^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {-\cot a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds -\frac {\cot a x... |
3 | 3 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosecant_of_a_x_over_Cotangent_of_a_x | $\ds \int \frac {\csc^2 a x \rd x} {\cot a x} = \frac {-\ln \size {\cot a x} } a + C$ | \(\ds \frac \d {\d x} \cot x\) \(=\) \(\ds -\csc^2 x\) Derivative of Cotangent Function \(\ds \leadsto \ \ \) \(\ds \int \frac {\csc^2 x \rd x} {\cot x}\) \(=\) \(\ds -\ln \size {\cot a x} + C\) Primitive of Function under its Derivative \(\ds \leadsto \ \ \) \(\ds ... |
4 | 4 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosine_Function | $\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$
where $C$ is an arbitrary constant .
Corollary
$\ds \int \cos^2 x \rd x = \frac {x + \sin x \cos x} 2 + C$ | \(\ds \int \cos^2 x \rd x\) \(=\) \(\ds \int \paren {\frac {1 + \cos 2 x} 2} \rd x\) Square of Cosine \(\ds \) \(=\) \(\ds \int \frac 1 2 \rd x + \int \paren {\frac {\cos 2 x} 2} \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \frac x 2 + C +... |
5 | 5 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosine_Function/Proof_1 | $\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | \(\ds \int \cos^2 x \rd x\) \(=\) \(\ds \int \paren {\frac {1 + \cos 2 x} 2} \rd x\) Square of Cosine \(\ds \) \(=\) \(\ds \int \frac 1 2 \rd x + \int \paren {\frac {\cos 2 x} 2} \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \frac x 2 + C +... |
6 | 6 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosine_Function/Proof_2 | $\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | \(\ds I_n\) \(=\) \(\ds \int \cos^n x \rd x\) \(\ds \) \(=\) \(\ds \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n-2}\) Reduction Formula for Integral of Power of Cosine \(\ds I_0\) \(=\) \(\ds \int \left({\cos x}\right)^0 \rd x\) ... |
7 | 7 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosine_Function/Proof_3 | $\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | \(\ds \int \cos^2 x \rd x\) \(=\) \(\ds \frac 1 4 \int \paren {e^{i x} + e^{-i x} }^2 \rd x\) Euler's Cosine Identity \(\ds \) \(=\) \(\ds \frac 1 4 \int \paren {e^{2 i x} + 2 + e^{-2 i x} } \rd x\) \(\ds \) \(=\) \(\ds \frac 1 4 \paren {\frac{e^{2 i x} - ... |
8 | 8 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosine_of_a_x | $\ds \int \cos^2 a x \rd x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$ | \(\ds \int \cos^2 x \rd x\) \(=\) \(\ds \frac x 2 + \frac {\sin 2 x} 4 + C\) Primitive of $\cos^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \cos^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\frac {a x} 2 + \frac {\sin 2 a x} 4} + C\) Primitive of Function of Constant Multiple ... |
9 | 9 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cosine_of_a_x_over_Sine_of_a_x | $\ds \int \frac {\cos^2 a x \rd x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \size {\tan \frac {a x} 2} + C$ | \(\ds \int \frac {\cos^2 a x \rd x} {\sin a x}\) \(=\) \(\ds \int \frac {\paren {1 - \sin^2 a x} \rd x} {\sin a x}\) Sum of Squares of Sine and Cosine \(\ds \) \(=\) \(\ds \int \frac {\d x} {\sin a x} - \int \sin a x \rd x\) Linear Combination of Primitives \(\ds ... |
10 | 10 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cotangent_Function | $\ds \int \cot^2 x \rd x = -\cot x - x + C$
where $C$ is an arbitrary constant . | \(\ds \int \cot^2 x \rd x\) \(=\) \(\ds \int \paren {\csc^2 x - 1} \rd x\) Difference of Squares of Cosecant and Cotangent \(\ds \) \(=\) \(\ds \int \csc^2 x \rd x + \int \paren {-1} \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds -\cot x + C... |
11 | 11 | https://proofwiki.org//wiki/Primitive_of_Square_of_Cotangent_of_a_x | $\ds \int \cot^2 a x \rd x = \frac {-\cot a x} a - x + C$ | \(\ds \int \cot^2 x \rd x\) \(=\) \(\ds -\cot x - x\) Primitive of $\cot^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \cot a x \rd x\) \(=\) \(\ds \frac 1 a \paren {-\cot a x - a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds \frac {-\cot... |
12 | 12 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_Function | $\ds \int \csch^2 x \rd x = -\coth x + C$
where $C$ is an arbitrary constant . | From Derivative of Hyperbolic Cotangent :
$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x$
The result follows from the definition of primitive .
$\blacksquare$ |
13 | 13 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_of_a_x | $\ds \int \csch^2 a x \rd x = \frac {-\coth a x} a + C$ | \(\ds \int \csch^2 x \rd x\) \(=\) \(\ds -\coth x + C\) Primitive of $\csch^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \csch^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {-\coth a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds \frac {-\co... |
14 | 14 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_of_a_x_over_Hyperbolic_Cotangent_of_a_x | $\ds \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$ | \(\ds \frac \d {\d x} \coth a x\) \(=\) \(\ds -a \csch^2 a x\) Derivative of $\coth a x$ \(\ds \leadsto \ \ \) \(\ds \int \frac {-a \csch^2 a x \rd x} {\coth a x}\) \(=\) \(\ds \ln \size {\coth a x} + C\) Primitive of Function under its Derivative \(\ds \leadsto \ \ \) ... |
15 | 15 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cosine_Function | $\ds \int \cosh^2 x \rd x = \frac {\sinh 2 x} 4 + \frac x 2 + C$
where $C$ is an arbitrary constant .
Corollary
$\ds \int \cosh^2 x \rd x = \frac {\sinh x \cosh x + x} 2 + C$ | \(\ds \int \cosh^2 x \rd x\) \(=\) \(\ds \int \paren {\frac {\cosh 2 x + 1} 2} \rd x\) Square of Hyperbolic Cosine \(\ds \) \(=\) \(\ds \int \paren {\frac {\cosh 2 x} 2} \rd x + \int \frac 1 2 \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \... |
16 | 16 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x | $\ds \int \cosh^2 a x \rd x = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C$
Corollary
$\ds \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$ | \(\ds \int \cosh^2 x \rd x\) \(=\) \(\ds \frac {\sinh x \cosh x + x} 2 + C\) Primitive of $\cosh^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \cosh^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\frac {\sinh a x \cosh a x + a x} 2} + C\) Primitive of Function of Constant Multiple ... |
17 | 17 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x/Corollary | $\ds \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$ | \(\ds \int \cosh^2 a x \rd x\) \(=\) \(\ds \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C\) Primitive of $\cosh^2 a x$ \(\ds \) \(=\) \(\ds \frac {\frac {\sinh 2 a x} 2} {2 a} + \frac x 2 + C\) Double Angle Formula for Hyperbolic Sine \(\ds \) \(=\) \(\... |
18 | 18 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x_over_Hyperbolic_Sine_of_a_x | $\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x} = \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$ | \(\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x}\) \(=\) \(\ds \int \frac {\paren {\sinh^2 a x + 1} \rd x} {\sinh a x}\) Difference of Squares of Hyperbolic Cosine and Sine \(\ds \) \(=\) \(\ds \int \frac {\sinh^2 a x \rd x} {\sinh a x} + \int \frac {\d x} {\sinh a x}\) Linear C... |
19 | 19 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cotangent_Function | $\ds \int \coth^2 x \rd x = x - \coth x + C$
where $C$ is an arbitrary constant . | \(\ds \int \coth^2 x \rd x\) \(=\) \(\ds \int \paren {1 + \csch^2 x} \rd x\) Difference of Squares of Hyperbolic Cotangent and Cosecant \(\ds \) \(=\) \(\ds \int 1 \rd x + \int \csch^2 x \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \int 1 ... |
20 | 20 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Cotangent_of_a_x | $\ds \int \coth^2 a x \rd x = x - \frac {\coth a x} a + C$ | \(\ds \int \coth^2 x \rd x\) \(=\) \(\ds x - \coth x + C\) Primitive of $\coth^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \coth^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {a x - \coth a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds x -... |
21 | 21 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Secant_Function | $\ds \int \sech^2 x \rd x = \tanh x + C$
where $C$ is an arbitrary constant . | From Derivative of Hyperbolic Tangent :
$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x$
The result follows from the definition of primitive .
$\blacksquare$ |
22 | 22 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Secant_of_a_x | $\ds \int \sech^2 a x \rd x = \frac {\tanh a x} a + C$ | \(\ds \int \sech^2 x \rd x\) \(=\) \(\ds \tanh x + C\) Primitive of $\sech^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \sech^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\tanh a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds \frac {\tanh ... |
23 | 23 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Secant_of_a_x_over_Hyperbolic_Tangent_of_a_x | $\ds \int \frac {\sech^2 a x \rd x} {\tanh a x} = \frac 1 a \ln \size {\tanh a x} + C$ | \(\ds \frac {\d} {\d x} \tanh a x\) \(=\) \(\ds a \sech^2 a x\) Derivative of $\tanh a x$ \(\ds \leadsto \ \ \) \(\ds \int \frac {a \sech^2 a x \rd x} {\tanh a x}\) \(=\) \(\ds \ln \size {\tanh a x} + C\) Primitive of Function under its Derivative \(\ds \leadsto \ \ \) ... |
24 | 24 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Sine_Function | $\ds \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$
where $C$ is an arbitrary constant .
Corollary
$\ds \int \sinh^2 x \rd x = \frac {\sinh x \cosh x - x} 2 + C$ | \(\ds \int \sinh^2 x \rd x\) \(=\) \(\ds \int \paren {\frac {\cosh 2 x - 1} 2} \rd x\) Square of Hyperbolic Sine \(\ds \) \(=\) \(\ds \int \paren {\frac {\cosh 2 x} 2} \rd x - \int \frac 1 2 \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \in... |
25 | 25 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x | $\ds \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$
Corollary
$\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$ | \(\ds \int \sinh^2 x \rd x\) \(=\) \(\ds \frac {\sinh x \cosh x - x} 2 + C\) Primitive of $\sinh^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \sinh^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\frac {\sinh a x \cosh a x - a x} 2} + C\) Primitive of Function of Constant Multiple ... |
26 | 26 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x/Corollary | $\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$ | \(\ds \int \sinh^2 a x \rd x\) \(=\) \(\ds \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C\) Primitive of $\sinh^2 a x$ \(\ds \) \(=\) \(\ds \frac {\frac {\sinh 2 a x} 2} {2 a} - \frac x 2 + C\) Double Angle Formula for Hyperbolic Sine \(\ds \) \(=\) \(\... |
27 | 27 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x_by_Square_of_Hyperbolic_Cosine_of_a_x | $\ds \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$ | \(\ds \int \sinh^2 a x \cosh^2 a x \rd x\) \(=\) \(\ds \int \paren {\sinh a x \cosh a x}^2 \rd x\) \(\ds \) \(=\) \(\ds \int \paren {\frac {\sinh 2 a x} 2}^2 \rd x\) Double Angle Formula for Hyperbolic Sine \(\ds \) \(=\) \(\ds \frac 1 4 \int \sinh^2 2 a x... |
28 | 28 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x_over_Hyperbolic_Cosine_of_a_x | $\ds \int \frac {\sinh^2 a x \rd x} {\cosh a x} = \frac {\sinh a x} a - \frac 1 a \map \arctan {\sinh a x} + C$ | \(\ds \int \frac {\sinh^2 a x \rd x} {\cosh a x}\) \(=\) \(\ds \int \frac {\paren {\cosh^2 a x - 1} \rd x} {\cosh a x}\) Difference of Squares of Hyperbolic Cosine and Sine \(\ds \) \(=\) \(\ds \int \frac {\paren {\cosh^2 a x} \rd x} {\cosh a x} - \int \frac {\d x} {\cosh a x}\) ... |
29 | 29 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Tangent_Function | $\ds \int \tanh^2 x \rd x = x - \tanh x + C$
where $C$ is an arbitrary constant . | \(\ds \int \tanh^2 x \rd x\) \(=\) \(\ds \int \paren {1 - \sech^2 x} \rd x\) Sum of Squares of Hyperbolic Secant and Tangent \(\ds \) \(=\) \(\ds \int 1 \rd x - \int \sech^2 x \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \int 1 \rd x - \ta... |
30 | 30 | https://proofwiki.org//wiki/Primitive_of_Square_of_Hyperbolic_Tangent_of_a_x | $\ds \int \tanh^2 a x \rd x = x - \frac {\tanh a x} a + C$ | \(\ds \int \tanh^2 x \rd x\) \(=\) \(\ds x - \tanh x + C\) Primitive of $\tanh^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \tanh^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {a x - \tanh a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds x -... |
31 | 31 | https://proofwiki.org//wiki/Primitive_of_Square_of_Logarithm_of_x | $\ds \int \ln^2 x \rd x = x \ln^2 x - 2 x \ln x + 2 x + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \ln x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 x\) Derivative of $\ln x$
and let:
\(\ds \... |
32 | 32 | https://proofwiki.org//wiki/Primitive_of_Square_of_Secant_Function | $\ds \int \sec^2 x \rd x = \tan x + C$
where $C$ is an arbitrary constant . | From Derivative of Tangent Function :
$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x$
The result follows from the definition of primitive .
$\blacksquare$ |
33 | 33 | https://proofwiki.org//wiki/Primitive_of_Square_of_Secant_of_a_x | $\ds \int \sec^2 a x \rd x = \frac {\tan a x} a + C$ | \(\ds \int \sec^2 x \rd x\) \(=\) \(\ds \tan x + C\) Primitive of $\sec^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \sec^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\tan a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds \frac {\tan a x} a... |
34 | 34 | https://proofwiki.org//wiki/Primitive_of_Square_of_Secant_of_a_x_over_Tangent_of_a_x | $\ds \int \frac {\sec^2 a x \rd x} {\tan a x} = \frac 1 a \ln \size {\tan a x} + C$ | \(\ds \frac {\d} {\d x} \tan x\) \(=\) \(\ds \sec^2 x\) Derivative of Tangent Function \(\ds \leadsto \ \ \) \(\ds \int \frac {\sec^2 x \rd x} {\tan x}\) \(=\) \(\ds \ln \size {\tan a x} + C\) Primitive of Function under its Derivative \(\ds \leadsto \ \ \) \(\ds \i... |
35 | 35 | https://proofwiki.org//wiki/Primitive_of_Square_of_Sine_Function | $\ds \int \sin^2 x \rd x = \frac x 2 - \frac {\sin 2 x} 4 + C$
where $C$ is an arbitrary constant .
Corollary
$\ds \int \sin^2 x \rd x = \frac {x - \sin x \cos x} 2 + C$ | \(\ds \int \sin^2 x \rd x\) \(=\) \(\ds \int \paren {\frac {1 - \cos 2 x} 2} \rd x\) Square of Sine \(\ds \) \(=\) \(\ds \frac 1 2 \int \d x - \frac 1 2 \int \cos 2 x \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \frac x 2 - \frac 1 2 \int ... |
36 | 36 | https://proofwiki.org//wiki/Primitive_of_Square_of_Sine_of_a_x | $\ds \int \sin^2 a x \rd x = \frac x 2 - \frac {\sin 2 a x} {4 a} + C$ | \(\ds \int \sin^2 x \rd x\) \(=\) \(\ds \frac x 2 - \frac {\sin 2 x} 4 + C\) Primitive of $\sin^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \sin^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\frac {a x} 2 - \frac {\sin 2 a x} 4} + C\) Primitive of Function of Constant Multiple ... |
37 | 37 | https://proofwiki.org//wiki/Primitive_of_Square_of_Sine_of_a_x_over_Cosine_of_a_x | $\ds \int \frac {\sin^2 a x \rd x} {\cos a x} = \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ | \(\ds \int \frac {\sin^2 a x \rd x} {\cos a x}\) \(=\) \(\ds \int \frac {\paren {1 - \cos^2 a x} \rd x} {\cos a x}\) Sum of Squares of Sine and Cosine \(\ds \) \(=\) \(\ds \int \frac {\d x} {\cos a x} - \int \cos a x \rd x\) Linear Combination of Primitives \(\ds ... |
38 | 38 | https://proofwiki.org//wiki/Primitive_of_Square_of_Tangent_Function | $\ds \int \tan^2 x \rd x = \tan x - x + C$
where $C$ is an arbitrary constant . | \(\ds \int \tan^2 x \rd x\) \(=\) \(\ds \int \paren {\sec^2 x - 1} \rd x\) Difference of Squares of Secant and Tangent \(\ds \) \(=\) \(\ds \int \sec^2 x \rd x + \int \paren {-1} \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \tan x + C + \i... |
39 | 39 | https://proofwiki.org//wiki/Primitive_of_Square_of_Tangent_Function/Proof_1 | $\ds \int \tan^2 x \rd x = \tan x - x + C$ | \(\ds \int \tan^2 x \rd x\) \(=\) \(\ds \int \paren {\sec^2 x - 1} \rd x\) Difference of Squares of Secant and Tangent \(\ds \) \(=\) \(\ds \int \sec^2 x \rd x + \int \paren {-1} \rd x\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \tan x + C + \i... |
40 | 40 | https://proofwiki.org//wiki/Primitive_of_Square_of_Tangent_Function/Proof_2 | $\ds \int \tan^2 x \rd x = \tan x - x + C$ | \(\ds I_n\) \(=\) \(\ds \int \tan^n x \rd x\) \(\ds \) \(=\) \(\ds \frac {\tan^{n - 1} x} {n - 1} - I_{n - 2}\) Reduction Formula for Integral of Power of Tangent \(\ds I_0\) \(=\) \(\ds \int \paren {\tan x}^0 \rd x\) \(\ds \) \(=\) ... |
41 | 41 | https://proofwiki.org//wiki/Primitive_of_Square_of_Tangent_of_a_x | $\ds \int \tan^2 a x \rd x = \frac {\tan a x} a - x + C$ | \(\ds \int \tan^2 x \rd x\) \(=\) \(\ds \tan x - x + C\) Primitive of $\tan^2 x$ \(\ds \leadsto \ \ \) \(\ds \int \tan^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\tan a x - a x} + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) \(\ds \frac {\... |
42 | 42 | https://proofwiki.org//wiki/Primitive_of_Tangent_Function/Cosine_Form | $\ds \int \tan x \rd x = -\ln \size {\cos x} + C$
where $\cos x \ne 0$. | \(\ds \int \tan x \rd x\) \(=\) \(\ds \int \frac {\sin x} {\cos x} \rd x\) Definition of Real Tangent Function \(\ds \) \(=\) \(\ds -\int \frac {-\sin x} {\cos x} \rd x\) Primitive of Constant Multiple of Function \(\ds \) \(=\) \(\ds -\int \frac {\paren {... |
43 | 43 | https://proofwiki.org//wiki/Primitive_of_Tangent_Function/Cosine_Form/Proof | $\ds \int \tan x \rd x = -\ln \size {\cos x} + C$
where $\cos x \ne 0$. | \(\ds \int \tan x \rd x\) \(=\) \(\ds \int \frac {\sin x} {\cos x} \rd x\) Definition of Real Tangent Function \(\ds \) \(=\) \(\ds -\int \frac {-\sin x} {\cos x} \rd x\) Primitive of Constant Multiple of Function \(\ds \) \(=\) \(\ds -\int \frac {\paren {... |
44 | 44 | https://proofwiki.org//wiki/Primitive_of_Tangent_Function/Secant_Form | $\ds \int \tan x \rd x = \ln \size {\sec x} + C$
where $\sec x$ is defined. | \(\ds \int \tan x \rd x\) \(=\) \(\ds -\ln \size {\cos x} + C\) Primitive of $\tan x$: Cosine Form \(\ds \) \(=\) \(\ds \ln \size {\frac 1 {\cos x} } + C\) Logarithm of Reciprocal \(\ds \) \(=\) \(\ds \ln \size {\sec x} + C\) Secant is Reciprocal of C... |
45 | 45 | https://proofwiki.org//wiki/Primitive_of_Tangent_Function/Secant_Form/Proof | $\ds \int \tan x \rd x = \ln \size {\sec x} + C$
where $\sec x$ is defined. | \(\ds \int \tan x \rd x\) \(=\) \(\ds -\ln \size {\cos x} + C\) Primitive of $\tan x$: Cosine Form \(\ds \) \(=\) \(\ds \ln \size {\frac 1 {\cos x} } + C\) Logarithm of Reciprocal \(\ds \) \(=\) \(\ds \ln \size {\sec x} + C\) Secant is Reciprocal of C... |
46 | 46 | https://proofwiki.org//wiki/Primitive_of_Tangent_of_a_x/Cosine_Form | $\ds \int \tan a x \rd x = \frac {-\ln \size {\cos a x} } a + C$ | \(\ds \int \tan x \rd x\) \(=\) \(\ds -\ln \size {\cos x}\) Primitive of $\tan x$: Cosine Form \(\ds \leadsto \ \ \) \(\ds \int \tan a x \rd x\) \(=\) \(\ds \frac 1 a \paren {-\ln \size {\cos a x} } + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) ... |
47 | 47 | https://proofwiki.org//wiki/Primitive_of_Tangent_of_a_x/Secant_Form | $\ds \int \tan a x \rd x = \frac {\ln \size {\sec a x} } a + C$ | \(\ds \int \tan x \rd x\) \(=\) \(\ds \ln \size {\sec x}\) Primitive of $\tan x$: Secant Form \(\ds \leadsto \ \ \) \(\ds \int \tan a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\ln \size {\sec a x} } + C\) Primitive of Function of Constant Multiple \(\ds \) \(=\) ... |
48 | 48 | https://proofwiki.org//wiki/Primitive_of_Tangent_of_a_x_over_x | $\ds \int \frac {\tan a x} x \rd x = a x + \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
where $B_n$ denotes the $n$th Bernoulli number . | \(\ds \int \frac {\tan a x} x \rd x\) \(=\) \(\ds \int \frac 1 x \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x\) Power Series Expansion for Tangent Function \(\ds \) \(=\) \(\ds \sum_{n \math... |
49 | 49 | https://proofwiki.org//wiki/Primitive_of_a_x_%2B_b_over_p_x_%2B_q | $\ds \int \frac {a x + b} {p x + q} \rd x = \frac {a x} p + \frac {b p - a q} {p^2} \ln \size {p x + q} + C$ | \(\ds \int \frac {a x + b} {p x + q} \rd x\) \(=\) \(\ds a \int \frac {x \rd x} {p x + q} + b \int \frac {\d x} {p x + q}\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds a \paren {\frac x p - \frac q {p^2} \ln \size {p x + q} } + b \int \frac {\d x} {p x + q} + C\) Primi... |
50 | 50 | https://proofwiki.org//wiki/Primitive_of_exp_(-x%5E2)_has_no_Solution_in_Elementary_Functions | The primitive :
$\ds \int \map \exp {-x^2} \rd x$
cannot be expressed in terms of a finite number of elementary functions . | This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof . To discuss this page in more detail, feel free to use the talk page . When this work has been completed, you may remove this instance of {{ ProofWanted }} from the code. If you would welcome a second opinion as to... |
51 | 51 | https://proofwiki.org//wiki/Primitive_of_exp_x_over_x_has_no_Solution_in_Elementary_Functions | The primitive :
$\ds \int \frac {e^{a x} \rd x} x$
cannot be expressed in terms of a finite number of elementary functions . | This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof . To discuss this page in more detail, feel free to use the talk page . When this work has been completed, you may remove this instance of {{ ProofWanted }} from the code. If you would welcome a second opinion as to... |
52 | 52 | https://proofwiki.org//wiki/Primitive_of_p_x_%2B_q_over_Root_of_a_x_%2B_b | $\ds \int \frac {p x + q} {\sqrt {a x + b} } \rd x = \frac {2 \paren {a p x + 3 a q - 2 b p} } {3 a^2} \sqrt{a x + b}$ | Let:
\(\ds u\) \(=\) \(\ds \sqrt{a x + b}\) \(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \frac {u^2 - b} a\)
Then:
\(\ds \int \frac {p x + q} {\sqrt {a x + b} } \rd x\) \(=\) \(\ds \int \paren {p \paren {\frac {u^2 - b} a} + q} \frac {2 u} {a u} \rd x\) Primitive of Functi... |
53 | 53 | https://proofwiki.org//wiki/Primitive_of_p_x_%2B_q_over_Root_of_a_x_squared_plus_2_b_x_plus_c | $\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x = \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }$ | \(\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x\) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x\) \(\ds \) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + 2 b - 2 b + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \r... |
54 | 54 | https://proofwiki.org//wiki/Primitive_of_p_x_%2B_q_over_Root_of_a_x_squared_plus_2_b_x_plus_c/Proof_1 | $\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x = \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }$ | \(\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x\) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x\) \(\ds \) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + 2 b - 2 b + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \r... |
55 | 55 | https://proofwiki.org//wiki/Primitive_of_p_x_%2B_q_over_Root_of_a_x_squared_plus_2_b_x_plus_c/Proof_2 | $\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x = \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }$ | \(\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x\) \(=\) \(\ds p \int \dfrac x {\sqrt {a x^2 + 2 b x + c} } \rd x + q \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds p \paren {\frac {\sqrt {a x^2 + 2 b x + c} } a -... |
56 | 56 | https://proofwiki.org//wiki/Primitive_of_p_x_%2B_q_over_a_x_squared_plus_2_b_x_plus_c | $\ds \int \dfrac {p x + q} {a x^2 + 2 b x + c} \rd x = \dfrac p {2 a} \ln \size {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {a^2 + 2 b x + c} + C$ | \(\ds \int \dfrac {p x + q} {a x^2 + 2 b x + c} \rd x\) \(=\) \(\ds p \int \dfrac x {a x^2 + 2 b x + c} \rd x + q \int \dfrac {\rd x} {a x^2 + 2 b x + c}\) Linear Combination of Primitives \(\ds \) \(=\) \(\ds \frac p {2 a} \ln \size {a x^2 + 2 b x + c} - \frac {p b} a \int \frac {\d... |
57 | 57 | https://proofwiki.org//wiki/Primitive_of_x_by_Arccosecant_of_x_over_a | $\ds \int x \arccsc \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arccsc \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arccsc \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end{cases}$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arccsc \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \begin {cases} \dfrac {-a} {x \sqrt {x^2 - ... |
58 | 58 | https://proofwiki.org//wiki/Primitive_of_x_by_Arccosine_of_x_over_a | $\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
\(\ds u\) \(=\) \(\ds \arccos \frac x a\) \(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \cos u\) \(=\) \(\ds \frac x a\) Definition of Real Arccosine \(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sin u\) \(=\) \(\ds \sqrt {1 - \frac {x^2} {a^2}... |
59 | 59 | https://proofwiki.org//wiki/Primitive_of_x_by_Arccosine_of_x_over_a/Proof_1 | $\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
\(\ds u\) \(=\) \(\ds \arccos \frac x a\) \(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \cos u\) \(=\) \(\ds \frac x a\) Definition of Real Arccosine \(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sin u\) \(=\) \(\ds \sqrt {1 - \frac {x^2} {a^2}... |
60 | 60 | https://proofwiki.org//wiki/Primitive_of_x_by_Arccosine_of_x_over_a/Proof_2 | $\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arccos \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-1} {\sqrt {a^2 - x^2} }\) Deri... |
61 | 61 | https://proofwiki.org//wiki/Primitive_of_x_by_Arccotangent_of_x_over_a | $\ds \int x \arccot \frac x a \rd x = \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arccot \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-a} {x^2 + a^2}\) Derivative of... |
62 | 62 | https://proofwiki.org//wiki/Primitive_of_x_by_Arcsecant_of_x_over_a | $\ds \int x \arcsec \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arcsec \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arcsec \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end{cases}$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arcsec \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \begin {cases} \dfrac a {x \sqrt {x^2 - a^2... |
63 | 63 | https://proofwiki.org//wiki/Primitive_of_x_by_Arcsine_of_x_over_a | $\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
\(\ds u\) \(=\) \(\ds \arcsin \frac x a\) \(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sin u\) \(=\) \(\ds \frac x a\) Definition of Real Arcsine \(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \cos u\) \(=\) \(\ds \sqrt {1 - \frac {x^2} {a^2} }... |
64 | 64 | https://proofwiki.org//wiki/Primitive_of_x_by_Arcsine_of_x_over_a/Proof_1 | $\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
\(\ds u\) \(=\) \(\ds \arcsin \frac x a\) \(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sin u\) \(=\) \(\ds \frac x a\) Definition of Real Arcsine \(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \cos u\) \(=\) \(\ds \sqrt {1 - \frac {x^2} {a^2} }... |
65 | 65 | https://proofwiki.org//wiki/Primitive_of_x_by_Arcsine_of_x_over_a/Proof_2 | $\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arcsin \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 {\sqrt {a^2 - x^2} }\) Derivat... |
66 | 66 | https://proofwiki.org//wiki/Primitive_of_x_by_Arctangent_of_x | $\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$ | From Primitive of $x \arctan \dfrac x a$ :
$\ds \int x \arctan \frac x a \rd x = \frac {x^2 + a^2} 2 \arctan \frac x a - \frac {a x} 2 + C$
The result follows on setting $a = 1$.
$\blacksquare$ |
67 | 67 | https://proofwiki.org//wiki/Primitive_of_x_by_Arctangent_of_x/Proof_1 | $\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$ | From Primitive of $x \arctan \dfrac x a$ :
$\ds \int x \arctan \frac x a \rd x = \frac {x^2 + a^2} 2 \arctan \frac x a - \frac {a x} 2 + C$
The result follows on setting $a = 1$.
$\blacksquare$ |
68 | 68 | https://proofwiki.org//wiki/Primitive_of_x_by_Arctangent_of_x/Proof_2 | $\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arctan x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 {x^2 + 1}\) Derivative of $\arctan x$
... |
69 | 69 | https://proofwiki.org//wiki/Primitive_of_x_by_Arctangent_of_x_over_a | $\ds \int x \arctan \frac x a \rd x = \frac {x^2 + a^2} 2 \arctan \frac x a - \frac {a x} 2 + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arctan \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac a {x^2 + a^2}\) Derivative of $\... |
70 | 70 | https://proofwiki.org//wiki/Primitive_of_x_by_Cosecant_of_a_x | $\ds \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ is the $2 n$th Bernoulli number . | \(\ds \int x \csc a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \int \theta \csc \theta \rd \theta\) Substitution of $a x \to \theta$ \(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \, \theta^{2 n - 1} }... |
71 | 71 | https://proofwiki.org//wiki/Primitive_of_x_by_Cosine_of_a_x | $\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
where $C$ is an arbitrary constant . | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function
and let:
\(\ds \fr... |
72 | 72 | https://proofwiki.org//wiki/Primitive_of_x_by_Cosine_of_x | $\ds \int x \cos x \rd x = \cos x + x \sin x + C$
where $C$ is an arbitrary constant . | From Primitive of $x \cos a x$ :
$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
The result follows on setting $a = 1$.
$\blacksquare$ |
73 | 73 | https://proofwiki.org//wiki/Primitive_of_x_by_Cosine_of_x/Proof_1 | $\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | From Primitive of $x \cos a x$ :
$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
The result follows on setting $a = 1$.
$\blacksquare$ |
74 | 74 | https://proofwiki.org//wiki/Primitive_of_x_by_Cosine_of_x/Proof_2 | $\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function
and let:
\(\ds \fr... |
75 | 75 | https://proofwiki.org//wiki/Primitive_of_x_by_Cosine_of_x/Proof_3 | $\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | We have:
\(\ds \map {\dfrac \d {\d x} } {\cos x + x \sin x}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\cos x} + \map {\dfrac \d {\d x} } {x \sin x}\) Sum Rule for Derivatives \(\ds \) \(=\) \(\ds \map {\dfrac \d {\d x} } {\cos x} + x \map {\dfrac \d {\d x} } {\sin x} + \map {\dfrac \d ... |
76 | 76 | https://proofwiki.org//wiki/Primitive_of_x_by_Cotangent_of_a_x | $\ds \int x \cot a x \rd x = \frac 1 {a ^ 2} \paren {a x - \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} - \cdots + \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1} !} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number . | From Power Series Expansion for Cotangent Function :
The (real) cotangent function has a Taylor series expansion :
\(\ds \cot x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}\) \(\ds \) \(=\) \(\ds \frac 1 x - \frac x 3... |
77 | 77 | https://proofwiki.org//wiki/Primitive_of_x_by_Exponential_of_a_x | $\ds \int x e^{a x} \rd x = \frac {e^{a x} } a \paren {x - \frac 1 a} + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function
and let:
\(\ds \fr... |
78 | 78 | https://proofwiki.org//wiki/Primitive_of_x_by_Exponential_of_a_x_by_Cosine_of_b_x | $\ds \int x e^{a x} \cos b x \rd x = \frac {x e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \cos b x - 2 a b \sin b x} } {\paren {a^2 + b^2}^2} + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function
and let:
\(\ds \fr... |
79 | 79 | https://proofwiki.org//wiki/Primitive_of_x_by_Exponential_of_a_x_by_Sine_of_b_x | $\ds \int x e^{a x} \sin b x \rd x = \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \sin b x - 2 a b \cos b x} } {\paren {a^2 + b^2}^2} + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function
and let:
\(\ds \fr... |
80 | 80 | https://proofwiki.org//wiki/Primitive_of_x_by_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | Let $a \in \R_{\ne 0}$.
Then:
$\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {a x^2 + b x + c}^{n + \frac 3 2} } {a \paren {2 n + 3} } - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x$ | \(\ds \) \(\) \(\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) \(\ds \) \(=\) \(\ds \int \frac {2 a x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x} {2 a}\) multiplying top and bottom by $2 a$ \(\ds \) \(=\) \(\ds \int \frac {\paren {2 a x + b... |
81 | 81 | https://proofwiki.org//wiki/Primitive_of_x_by_Hyperbolic_Cosecant_of_a_x | \(\ds \int x \csch a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C\) \(\ds \) \(=\) \(\ds \dfrac 1 {a^2} \paren {a x - \dfrac {\paren {a x}^3} {18} + \dfrac {7 \paren {a x}^5... | \(\ds \int x \csch a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \int \theta \csch \theta \rd \theta\) Substitution of $a x \to \theta$ \(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} ... |
82 | 82 | https://proofwiki.org//wiki/Primitive_of_x_by_Hyperbolic_Cosine_of_a_x | $\ds \int x \cosh a x \rd x = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C$
where $C$ is an arbitrary constant . | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
\(\ds u\) \(=\) \(\ds x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function
and let:
\(\ds \... |
83 | 83 | https://proofwiki.org//wiki/Primitive_of_x_by_Hyperbolic_Cotangent_of_a_x | $\ds \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number . | \(\ds \int x \coth a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \int \theta \coth \theta \rd \theta\) Substitution of $a x \to \theta$ \(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta\) Pow... |
84 | 84 | https://proofwiki.org//wiki/Primitive_of_x_by_Hyperbolic_Secant_of_a_x | \(\ds \int x \sech a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C\) \(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6... | \(\ds \int x \sech a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \int \theta \sech \theta \rd \theta\) Substitution of $a x \to \theta$ \(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{E_{2 n} \, \theta^{2 n} } {\paren {2 n}!} \rd \theta\) Power Series Ex... |
85 | 85 | https://proofwiki.org//wiki/Primitive_of_x_by_Hyperbolic_Sine_of_a_x | $\ds \int x \sinh a x \rd x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function
and let:
\(\ds \fr... |
86 | 86 | https://proofwiki.org//wiki/Primitive_of_x_by_Hyperbolic_Tangent_of_a_x | $\ds \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number . | \(\ds \int x \tanh a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \int \theta \tanh \theta \rd \theta\) Substitution of $a x \to \theta$ \(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} ... |
87 | 87 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosecant_of_x_over_a | $\ds \int x \arcsch \frac x a \rd x = \begin {cases}
\dfrac {x^2} 2 \arcsch \dfrac x a + \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x > 0 \\ \\
\dfrac {x^2} 2 \arcsch \dfrac x a - \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x < 0
\end {cases}$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arcsch \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \dfrac {-a} {\size x \sqrt {a^2 + x^2} }\) ... |
88 | 88 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a | $\ds \int x \arcosh \frac x a \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \arcosh \dfrac x a - \dfrac {x \sqrt {x^2 - a^2} } 4 + C$
where $\arcosh$ denotes the real area hyperbolic cosine .
Corollary
$\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \paren {-\cosh^{-1} \fr... | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arcosh \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 {\sqrt {x^2 - a^2} }\) Derivat... |
89 | 89 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary | $\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \paren {-\cosh^{-1} \frac x a} + \dfrac {x \sqrt {x^2 - a^2} } 4 + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction . | \(\ds -\cosh^{-1} \frac x a\) \(=\) \(\ds -\arcosh \frac x a\) Definition of Real Inverse Hyperbolic Cosine \(\ds \leadsto \ \ \) \(\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x\) \(=\) \(\ds -\int x \arcosh \frac x a \rd x\) \(\ds \) \(=\) \(\ds -\paren ... |
90 | 90 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cotangent_of_x_over_a | $\ds \int x \arcoth \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \arcoth \frac x a + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arcoth \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac a {a^2 - x^2}\) Derivative of $\... |
91 | 91 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a | $\ds \int x \arsech \frac x a \rd x = \dfrac {x^2} 2 \arsech \dfrac x a - \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $\arsech$ denotes the real area hyperbolic secant .
Corollary
$\ds \int x \paren {-\sech^{-1} \frac x a} \rd x = -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
wher... | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arsech \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-a} {x \sqrt {a^2 - x^2} }\) De... |
92 | 92 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary | $\ds \int x \paren {-\sech^{-1} \frac x a} \rd x = -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction . | \(\ds -\sech^{-1} \frac x a\) \(=\) \(\ds -\arsech \frac x a\) Definition of Real Inverse Hyperbolic Secant \(\ds \leadsto \ \ \) \(\ds \int x \paren {-\sech^{-1} \frac x a} \rd x\) \(=\) \(\ds -\int x \arsech \frac x a \rd x\) \(\ds \) \(=\) \(\ds -\paren ... |
93 | 93 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Sine_of_x_over_a | $\ds \int x \arsinh \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \arsinh \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \arsinh \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 {\sqrt {x^2 + a^2} }\) Derivat... |
94 | 94 | https://proofwiki.org//wiki/Primitive_of_x_by_Inverse_Hyperbolic_Tangent_of_x_over_a | $\ds \int x \artanh \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \artanh \frac x a + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \artanh \frac x a\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac a {a^2 - x^2}\) Derivative of $\... |
95 | 95 | https://proofwiki.org//wiki/Primitive_of_x_by_Logarithm_of_x | $\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \ln x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 x\) Derivative of $\ln x$
and let:
\(\ds \... |
96 | 96 | https://proofwiki.org//wiki/Primitive_of_x_by_Logarithm_of_x/Proof_1 | $\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | With a view to expressing the primitive in the form:
$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) \(=\) \(\ds \ln x\) \(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 x\) Derivative of $\ln x$
and let:
\(\ds \... |
97 | 97 | https://proofwiki.org//wiki/Primitive_of_x_by_Logarithm_of_x/Proof_2 | $\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | From Primitive of $x^m \ln x$ :
$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
The result follows by setting $m = 1$.
$\blacksquare$ |
98 | 98 | https://proofwiki.org//wiki/Primitive_of_x_by_Logarithm_of_x_squared_plus_a_squared | $\ds \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$ | \(\ds \int x \map \ln {x^2 + a^2} \rd x\) \(=\) \(\ds \frac {x^2 \map \ln {x^2 + a^2} } 2 - \int \frac {x^3} {x^2 + a^2} \rd x + C\) Primitive of $x^m \map \ln {x^2 + a^2}$ with $m = 1$ \(\ds \) \(=\) \(\ds \frac {x^2 \map \ln {x^2 + a^2} } 2 - \paren {\frac {x^2} 2 - \frac {a^2} 2 \m... |
99 | 99 | https://proofwiki.org//wiki/Primitive_of_x_by_Logarithm_of_x_squared_plus_a_squared/Proof_1 | $\ds \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$ | \(\ds \int x \map \ln {x^2 + a^2} \rd x\) \(=\) \(\ds \frac {x^2 \map \ln {x^2 + a^2} } 2 - \int \frac {x^3} {x^2 + a^2} \rd x + C\) Primitive of $x^m \map \ln {x^2 + a^2}$ with $m = 1$ \(\ds \) \(=\) \(\ds \frac {x^2 \map \ln {x^2 + a^2} } 2 - \paren {\frac {x^2} 2 - \frac {a^2} 2 \m... |
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