hcju/proofwikips huggingface dataset:
Primitive of Square of Secant of a x
:$\displaystyle \int \sec^2 a x \ \mathrm d x = \frac {\tan a x} a + C$
Primitive of Secant of a x/Tangent Form
:$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ where $\map \tan {\dfrac \pi 4 + \dfrac {a x} 2} \ne 0$.
Primitive of Secant of a x/Secant plus Tangent Form
:$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\sec a x + \tan a x} + C$ where $\sec a x + \tan a x \ne 0$.
Primitive of Power of Secant of a x
:$\displaystyle \int \sec^n a x \rd x = \frac {\sec^{n - 2} a x \tan a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \rd x$
Primitive of Reciprocal of Cube of Cosine of a x
:$\displaystyle \int \frac {\d x} {\cos^3 a x} = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
Primitive of Cosine of a x by Cosine of p x
:$\displaystyle \int \cos a x \cos p x \rd x = \frac {\map \sin {\paren {a - p} x} } {2 \paren {a - p} } + \frac {\map \sin {\paren {a + p} x} } {2 \paren {a + p} } + C$
Primitive of Reciprocal of 1 minus Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {1 - \cos a x} = \frac {-1} a \cot \frac {a x} 2 + C$
Primitive of x over 1 minus Cosine of a x
:$\displaystyle \int \frac {x \rd x} {1 - \cos a x} = \frac {-x} a \cot \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sin \frac {a x} 2} + C$
Primitive of Reciprocal of 1 plus Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {1 + \cos a x} = \frac 1 a \tan \frac {a x} 2 + C$
Primitive of Reciprocal of p plus q by Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {p + q \cos a x} = \begin{cases}  \displaystyle \frac 2 {a \sqrt {p^2 - q^2} } \arctan \left({\sqrt {\frac {p - q} {p + q} } \tan \dfrac {a x} 2}\right) + C & : p^2 > q^2 \\ \displaystyle \frac 1 {a \sqrt {q^2 - p^2} } \ln \left\vert{\frac {\tan \dfrac {a x} 2 + \sqrt {\dfrac {q + p} {q - p} } } {\tan \dfrac {a x} 2 - \sqrt {\dfrac {q + p} {q - p} } } }\right\vert + C & : p^2 < q^2 \\ \end{cases}$
Primitive of Power of x by Cosine of a x
:$\displaystyle \int x^m \cos a x \rd x = \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \cos a x \rd x$
Primitive of Reciprocal of Power of Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\cos^n a x} = \frac {\sin a x} {a \left({n - 1}\right) \cos^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\mathrm d x} {\cos^{n - 2} a x}$
Primitive of x over Power of Cosine of a x
:$\displaystyle \int \frac {x \ \mathrm d x} {\cos^n a x} = \frac {x \sin a x} {a \left({n - 1}\right) \cos^{n - 1} a x} - \frac 1 {a^2 \left({n - 1}\right) \left({n - 2}\right) \cos^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\mathrm d x} {\cos^{n - 2} a x} + C$
Primitive of Sine of a x by Cosine of a x
:$\displaystyle \int \sin a x \cos a x \rd x = \frac {\sin^2 a x} {2 a} + C$
Primitive of Sine of p x by Cosine of q x
:$\displaystyle \int \sin p x \cos q x \rd x = \frac {-\cos \left({p - q}\right) x} {2 \left({p - q}\right)} - \frac {\cos \left({p + q}\right) x} {2 \left({p + q}\right)} + C$ for $p, q \in \R: p \ne q$
Primitive of Power of Sine of a x by Cosine of a x
:$\displaystyle \int \sin^n a x \cos a x \rd x = \frac {\sin^{n + 1} a x} {\paren {n + 1} a} + C$
Primitive of Power of Cosine of a x by Sine of a x
:$\displaystyle \int \cos^n a x \sin a x \rd x = \frac {-\cos^{n + 1} a x} {\paren {n + 1} a} + C$
Primitive of Sine of a x squared by Cosine of a x squared
:$\displaystyle \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$
Primitive of Reciprocal of Sine of a x by Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sin a x \cos a x} = \frac 1 a \ln \left\vert{\tan a x}\right\vert + C$
Primitive of Square of Secant of a x over Tangent of a x
:$\displaystyle \int \frac {\sec^2 a x \ \mathrm d x} {\tan a x} = \frac 1 a \ln \left\vert{\tan a x}\right\vert + C$
Primitive of Reciprocal of Square of Sine of a x by Cosine of a x
:$\displaystyle \int \frac {\d x} {\sin^2 a x \cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } - \frac 1 {a \sin a x} + C$
Primitive of Power of Cosecant of a x by Cotangent of a x
:$\displaystyle \int \csc^n a x \cot a x \ \mathrm d x = \frac {-\csc^n a x} {n a} + C$
Primitive of Square of Sine of a x over Cosine of a x
:$\displaystyle \int \frac {\sin^2 a x \rd x} {\cos a x} = \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
Primitive of Square of Cosine of a x over Sine of a x
:$\displaystyle \int \frac {\cos^2 a x \ \mathrm d x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \left\vert{\tan \frac {a x} 2}\right\vert + C$
Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\cos a x \left({1 + \sin a x}\right)} = \frac {-1} {2 a \left({1 + \sin a x}\right)} + \frac 1 {2 a} \ln \left\vert{\tan \left({\frac {a x} 2 + \frac \pi 4}\right)}\right\vert + C$
Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x
:$\displaystyle \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$
Primitive of Reciprocal of Sine of a x by 1 plus Cosine of a x
:$\displaystyle \int \frac {\d x} {\sin a x \paren {1 + \cos a x} } = \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$
Primitive of Reciprocal of Sine of a x plus Cosine of a x
:$\displaystyle \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$
Primitive of Reciprocal of Sine of a x minus Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sin a x - \cos a x} = \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac {a x} 2 - \frac \pi 8}\right)}\right\vert + C$
Primitive of Sine of a x over Sine of a x plus Cosine of a x
:$\displaystyle \int \frac {\sin a x \ \mathrm d x} {\sin a x + \cos a x} = \frac x 2 - \frac 1 {2 a} \ln \left\vert{\sin a x + \cos a x}\right\vert + C$
Primitive of Sine of a x over Sine of a x minus Cosine of a x
:$\displaystyle \int \frac {\sin a x \rd x} {\sin a x - \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$
Primitive of Cosine of a x over Sine of a x plus Cosine of a x
:$\displaystyle \int \frac {\cos a x \rd x} {\sin a x + \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$
Primitive of Cosine of a x over Sine of a x minus Cosine of a x
:$\displaystyle \int \frac {\cos a x \ \mathrm d x} {\sin a x - \cos a x} = \frac {-x} 2 + \frac 1 {2 a} \ln \left\vert{\sin a x - \cos a x}\right\vert + C$
Primitive of Sine of a x over p plus q of Cosine of a x
:$\displaystyle \int \frac {\sin a x \rd x} {p + q \cos a x} = \frac {-1} {a q} \ln \size {p + q \cos a x} + C$
Primitive of Sine of a x over Power of p plus q of Cosine of a x
:$\displaystyle \int \frac {\sin a x \ \mathrm d x} {\left({p + q \cos a x}\right)^n} = \frac 1 {a q \left({n - 1}\right) \left({p + q \cos a x}\right)^{n - 1} } + C$
Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {p \sin a x + q \cos a x} = \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \left\vert{\frac {a x + \arctan \dfrac q p} 2}\right\vert + C$
Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus r
:$\ds \int \frac {\d x} {p \sin a x + q \cos a x + r} = \begin{cases} \ds \frac 2 {a \sqrt {r^2 - p^2 - q^2} } \map \arctan {\frac {p + \paren {r - q} \tan \dfrac {a x} 2} {\sqrt {r^2 - p^2 - q^2} } } + C & : p^2 + q^2 < r^2 \\ \ds \frac 1 {a \sqrt {p^2 + q^2 - r^2} } \ln \size {\frac {p - \sqrt {p^2 + q^2 - r^2} + \paren {r - q} \tan \dfrac {a x} 2} {p + \sqrt {p^2 + q^2 - r^2} + \paren {r - q} \tan \dfrac {a x} 2} } + C & : p^2 + q^2 > r^2 \end{cases}$
Primitive of Reciprocal of p by Sine of a x plus q by 1 plus Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {p \sin a x + q \left({1 + \cos a x}\right)} = \frac 1 {a p} \ln \left\vert{q + p \tan \frac {a x} 2}\right\vert + C$
Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x
:$\displaystyle \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x} = \frac 1 {a p q} \map \arctan {\frac {p \tan a x} q} + C$
Primitive of Reciprocal of p squared by square of Sine of a x minus q squared by square of Cosine of a x
:$\displaystyle \int \frac {\d x} {p^2 \sin^2 a x - q^2 \cos^2 a x} = \frac 1 {2 a p q} \ln \size {\frac {p \tan a x - q} {p \tan a x + q} } + C$
Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Sine
:$\displaystyle \int \sin^m a x \cos^n a x \rd x = \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$
Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Cosine
:$\displaystyle \int \sin^m a x \cos^n a x \rd x = \frac {\sin^{m + 1} a x \cos^{n - 1} a x} {a \paren {m + n} } + \frac {n - 1} {m + n} \int \sin^m a x \cos^{n - 2} a x \rd x + C$
Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Both Powers
:$\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m - 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\sin^{m - 2} a x} {\cos^{n - 2} a x} \rd x + C$
Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine
:$\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \ \mathrm d x = \frac {\sin^{m + 1} a x} {a \left({n - 1}\right) \cos^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \ \mathrm d x + C$
Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Both Powers
:$\displaystyle \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m - 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\cos^{m - 2} a x} {\sin^{n - 2} a x} \rd x + C$
Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine
:$\displaystyle \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C$
Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Cosine
:$\displaystyle \int \frac {\mathrm d x} {\sin^m a x \cos^n a x} = \frac 1 {a \left({n - 1}\right) \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\mathrm d x} {\sin^m a x \cos^{n - 2} a x}$
Primitive of Tangent of a x/Cosine Form
:$\ds \int \tan a x \rd x = \frac {-\ln \size {\cos a x} } a + C$
Primitive of Square of Tangent of a x
:$\displaystyle \int \tan^2 a x \ \mathrm d x = \frac {\tan a x} a - x + C$
Primitive of Cube of Tangent of a x
:$\displaystyle \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$
Primitive of Power of Tangent of a x by Square of Secant of a x
:$\displaystyle \int \tan^n a x \sec^2 a x \rd x = \frac {\tan^{n + 1} a x} {\paren {n + 1} a} + C$
Primitive of Cotangent of a x
:$\displaystyle \int \cot a x \rd x = \frac {\ln \size {\sin a x} } a + C$
Primitive of x by Tangent of a x
:$\displaystyle \int x \tan a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x} ^ 3} 3 + \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
Primitive of Tangent of a x over x
:$\displaystyle \int \frac {\tan a x} x \ \mathrm d x = a x + \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$ where $B_n$ denotes the $n$th Bernoulli number.
Primitive of Reciprocal of p plus q by Tangent of a x
:$\displaystyle \int \frac {\d x} {p + q \tan a x} = \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {q \sin a x + p \cos a x} + C$
Primitive of Square of Cotangent of a x
:$\displaystyle \int \cot^2 a x \rd x = \frac {-\cot a x} a - x + C$
Primitive of Cube of Cotangent of a x
:$\displaystyle \int \cot^3 a x \rd x = \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$
Primitive of Power of Cotangent of a x by Square of Cosecant of a x
:$\displaystyle \int \cot^n a x \csc^2 a x \ \mathrm d x = \frac {-\cot^{n + 1} a x} {\left({n + 1}\right) a} + C$
Primitive of Square of Cosecant of a x over Cotangent of a x
:$\displaystyle \int \frac {\csc^2 a x \ \mathrm d x} {\cot a x} = \frac {-\ln \left\vert{\cot a x}\right\vert} a + C$
Primitive of Cotangent of a x over x
:$\displaystyle \int \frac {\cot a x} x \ \mathrm d x = \frac {-1} a x - \frac {a x} 3 - \frac {\paren {a x}^3} {135} - \cdots - \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} - \cdots + C$ where $B_n$ denotes the $n$th Bernoulli number.
Primitive of x by Square of Cotangent of a x
:$\displaystyle \int x \cot^2 a x \ \mathrm d x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \left\vert{\sin a x}\right\vert - \frac {x^2} 2 + C$
Primitive of Power of Cotangent of a x
:$\displaystyle \int \cot^n a x \ \mathrm d x = \frac {-\cot^{n - 1} a x} {\left({n - 1}\right) a} - \int \cot^{n - 2} a x \ \mathrm d x + C$
Primitive of Cube of Secant of a x
:$\displaystyle \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$
Primitive of x by Secant of a x
{{begin-eqn}} {{eqn | l = \int x \sec a x \rd x       | r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots + \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + \cdots} + C       | c =  }} {{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C       | c =  }} {{end-eqn}} where $E_{2 n}$ is the $2 n$th Euler number.
Primitive of Secant of a x over x
:$\displaystyle \int \frac {\sec a x} x \ \mathrm d x = \ln \size x + \frac {\paren {a x}^2} 4 + \frac {5 \paren {a x}^4} {96} + \frac {61 \paren {a x}^6} {4320} + \cdots + \frac {\paren {-1}^n E_n \paren {a x}^{2 n} } {\paren {2 n} \paren {2 n}!} + \cdots + C$ where $E_n$ is the $n$th Euler number.
Primitive of x by Square of Secant of a x
:$\displaystyle \int x \sec^2 a x \ \mathrm d x = \frac {x \tan a x} a + \frac  1 {a^2} \ln \left\vert{\cos a x}\right\vert + C$
Primitive of Reciprocal of q plus p by Secant of a x
:$\ds \int \frac {\d x} {q + p \sec a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \cos a x} + C$
Primitive of x by Cosecant of a x
:$\displaystyle \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_n \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ where $B_{2 n}$ is the $2 n$th Bernoulli number.
Primitive of Cosecant of a x over x
:$\displaystyle \int \frac {\csc a x} x \rd x = \frac {-1} {a x} + \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$ where $B_n$ is the $n$th Bernoulli number.
Primitive of x by Square of Cosecant of a x
:$\displaystyle \int x \csc^2 a x \ \mathrm d x = \frac {-x \cot a x} a + \frac  1 {a^2} \ln \left\vert{\sin a x}\right\vert + C$
Primitive of Reciprocal of q plus p by Cosecant of a x
:$\displaystyle \int \frac {\mathrm d x} {q + p \csc a x} = \frac x q - \frac p q \int \frac {\mathrm d x} {p + q \sin a x} + C$
Primitive of Arcsine of x over a
:$\displaystyle \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$
Primitive of x squared by Arcsine of x over a
:$\displaystyle \int x^2 \arcsin \frac x a \rd x = \frac {x^3} 3 \arcsin \frac x a + \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$
Primitive of Arccosine of x over a
:$\displaystyle \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$
Primitive of x by Arccosine of x over a/Proof 1
:$\displaystyle \int x \arccos \frac x a \ \mathrm d x = \left({\frac {x^2} 2 - \frac {a^2} 4}\right) \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$
Primitive of x by Arccosine of x over a/Proof 2
:$\displaystyle \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$
Primitive of x squared by Arccosine of x over a
:$\displaystyle \int x^2 \arccos \frac x a \rd x = \frac {x^3} 3 \arccos \frac x a - \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$
Primitive of Arccosine of x over a over x squared
:$\displaystyle \int \frac {\arccos \frac x a \ \mathrm d x} {x^2} = \frac {-\arccos \frac x a} x + \frac 1 a \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C$
Primitive of Square of Arccosine of x over a
:$\displaystyle \int \paren {\arccos \frac x a}^2 \rd x = x \paren {\arccos \frac x a}^2 - 2 x - 2 \sqrt {a^2 - x^2} \arccos \frac x a + C$
Primitive of Arctangent of x over a
:$\displaystyle \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \ln \paren {x^2 + a^2} + C$
Primitive of x by Arctangent of x over a
:$\displaystyle \int x \arctan \frac x a \ \mathrm d x = \frac {x^2 + a^2} 2 \arctan \frac x a - \frac {a x} 2 + C$
Primitive of x by Arccotangent of x over a
:$\displaystyle \int x \arccot \frac x a \rd x = \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 + C$
Primitive of x squared by Arctangent of x over a
:$\displaystyle \int x^2 \arctan \frac x a \ \mathrm d x = \frac {x^3} 3 \arctan \frac x a - \frac {a x^2} 6 + \frac {a^3} 6 \ln \left({x^2 + a^2}\right) + C$
Primitive of Arctangent of x over a over x
{{begin-eqn}} {{eqn 	| l = \int \frac 1 x \arctan \paren {\frac x a} \rd x       	| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {x^{2 k + 1} } {\paren {2 k + 1}^2 a^{2 k + 1} }  }} {{eqn 	| r = \frac x a - \frac {x^3} {3^2 a^3} + \frac {x^5} {5^2 a^5} - \frac {x^7} {7^2 a^7} + \cdots + C }} {{end-eqn}}
Primitive of Arccotangent of x over a over x squared
:$\displaystyle \int \frac {\arccot \frac x a \rd x} {x^2} = \frac {-\arccot \frac x a} x + \frac 1 {2 a} \map \ln {\frac {x^2 + a^2} {x^2} } + C$
Primitive of Arccotangent of x over a
:$\displaystyle \int \operatorname{arccot} \frac x a \ \mathrm d x = x \operatorname{arccot} \frac x a + \frac a 2 \ln \left({x^2 + a^2}\right) + C$
Primitive of Arccotangent of x over a/Proof 1
:$\displaystyle \int \arccot \frac x a \rd x = x \arccot \frac x a + \frac a 2 \, \map \ln {x^2 + a^2} + C$
Primitive of Arccotangent of x over a/Proof 2
:$\displaystyle \int \operatorname{arccot} \frac x a \ \mathrm d x = x \operatorname{arccot} \frac x a + \frac a 2 \ln \left({x^2 + a^2}\right) + C$
Primitive of x by Arcsecant of x over a
:$\displaystyle \int x \operatorname{arcsec} \frac x a \ \mathrm d x = \begin{cases} \displaystyle \frac {x^2} 2 \operatorname{arcsec} \frac x a - \frac {a \sqrt{x^2 - a^2} } 2 + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^2} 2 \operatorname{arcsec} \frac x a + \frac {a \sqrt{x^2 - a^2} } 2 + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$
Primitive of x by Arccosecant of x over a
:$\displaystyle \int x \arccsc \frac x a \rd x = \begin{cases} \displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac {a \sqrt {x^2 - a^2} } 2 + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$
Primitive of x squared by Arcsecant of x over a
:$\displaystyle \int x^2 \operatorname{arcsec} \frac x a \ \mathrm d x = \begin{cases} \displaystyle \frac {x^3} 3 \operatorname{arcsec} \frac x a - \frac {a x \sqrt{x^2 - a^2} } 6 - \frac {a^3} 6 \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^3} 3 \operatorname{arcsec} \frac x a + \frac {a x \sqrt{x^2 - a^2} } 6 + \frac {a^3} 6 \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$
Primitive of Arccosecant of x over a over x squared
:$\displaystyle \int \frac {\arccsc \frac x a} {x^2} \rd x = \begin{cases} \displaystyle \frac {-\arccsc \frac x a} x - \frac {\sqrt{x^2 - a^2} } {a x} + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {-\arccsc \frac x a} x + \frac {\sqrt{x^2 - a^2} } {a x} + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$
Primitive of Power of x by Arccosecant of x over a
:$\displaystyle \int x^m \arccsc \frac x a \rd x = \begin{cases} \displaystyle \frac {x^{m + 1} } {m + 1} \arccsc \frac x a + \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} } + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$
Primitive of Exponential of a x
:$\displaystyle \int e^{a x} \rd x = \frac {e^{a x} } a + C$
Primitive of x by Exponential of a x
:$\displaystyle \int x e^{a x} \rd x = \frac {e^{a x} } a \paren {x - \frac 1 a} + C$
Primitive of x squared by Exponential of a x
:$\displaystyle \int x^2 e^{a x} \rd x = \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C$
Primitive of Power of x by Exponential of a x
Let $n$ be a positive integer. Let $a$ be a non-zero real number. Then: {{begin-eqn}} {{eqn | l = \int x^n e^{a x} \rd x       | r = \frac {e^{a x} } a \paren {x^n - \dfrac {n x^{n - 1} } a + \dfrac {n \paren {n - 1} x^{n - 2} } {a^2} - \dfrac {n \paren {n - 1} \paren {n - 2} x^{n - 3} } {a^3} + \cdots + \dfrac {\paren {-1}^n n!} {a^n} } + C       | c =  }} {{eqn | r = \frac {e^{a x} } a \sum_{k \mathop = 0}^n \paren {\paren {-1}^k \frac {n^{\underline k} x^{n - k} } {a^k} } + C       | c =  }} {{end-eqn}} where $n^{\underline k}$ denotes the $k$th falling factorial power of $n$.
Primitive of Power of x by Exponential of a x/Lemma
Let $n$ be a positive integer. Then: :$\displaystyle \int x^n e^{a x} \rd x = \frac {x^n e^{a x} } a - \frac n a \int x^{n - 1} e^{a x} \rd x + C$
Primitive of Exponential of a x over x
:$\displaystyle \int \frac {e^{a x} \rd x} x = \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C$
Primitive of Exponential of a x over Power of x
:$\displaystyle \int \frac {e^{a x} \rd x} {x^n} = \frac {-e^{a x} } {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \rd x} {x^{n - 1} } + C$ where $n \ne 1$.
Primitive of Reciprocal of p plus q by Exponential of a x
:$\displaystyle \int \frac {\d x} {p + q e^{a x} } = \frac x p - \frac 1 {a p} \ln \size {p + q e^{a x} } + C$
Primitive of Reciprocal of p by Exponential of a x plus q by Exponential of -a x
:$\displaystyle \int \frac {\d x} {p e^{a x} + q e^{-a x} } = \begin{cases} \displaystyle \frac 1 {a \sqrt {p q} } \map \arctan {\sqrt {\frac p q} e^{a x} } & : \sqrt {p q} > 0 \\ \displaystyle \frac 1 {2 a \sqrt {-p q} } \ln \size {\frac {e^{a x} - \sqrt {-\frac q p} } {e^{a x} + \sqrt {-\frac q p} } } & : \sqrt {p q} < 0 \\ \end{cases}$
Primitive of Exponential of a x by Sine of b x
:$\displaystyle \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$
Primitive of Exponential of a x by Cosine of b x
:$\displaystyle \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x}} {a^2 + b^2} + C$
Primitive of x by Exponential of a x by Sine of b x
:$\displaystyle \int x e^{a x} \sin b x \ \mathrm d x = \frac {x e^{a x} \left({a \sin b x - b \cos bx}\right)} {a^2 + b^2} - \frac {e^{a x} \left({\left({a^2 - b^2}\right) \sin b x - 2 a b \cos bx}\right)} {\left({a^2 + b^2}\right)^2} + C$
Primitive of Exponential of a x by Logarithm of x
:$\displaystyle \int e^{a x} \ln x \rd x = \frac {e^{a x} \ln x} a - \frac 1 a \int \frac {e^{a x} } x \rd x + C$