hcju/proofwikiqa huggingface dataset:
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^m       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = m x^{m - 1}       | c = Power Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \sin a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {- \cos a x} a       | c = Primitive of $\sin a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^m \sin a x \rd x       | r = \paren {x^m} \paren {\frac {-\cos a x} a} - \int \paren {\frac {-\cos a x} a} \paren {m x^{m - 1} } \rd x + C       | c = Integration by Parts }} {{eqn | n = 1       | r = \frac {- x^m \cos a x} a + \frac m a \int x^{m - 1} \cos a x \rd x + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} Similarly, let: {{begin-eqn}} {{eqn | l = u       | r = x^{m - 1}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \paren {m - 1} x^{m - 2}       | c = Power Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \cos a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\sin a x} a       | c = Primitive of $\cos a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^{m - 1} \cos a x \rd x       | r = \paren {x^{m - 1} } \paren {\frac {\sin a x} a} - \int \paren {\frac {\sin a x} a} \paren {m - 1} x^{m - 2} \rd x + C       | c = Integration by Parts }} {{eqn | n = 2       | r = \frac {x^{m - 1} \sin a x} a - \frac {m - 1} a \int x^{m - 2} \sin a x \rd x + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} So: {{begin-eqn}} {{eqn | l = \int x^m \sin a x \rd x       | r = \frac {- x^m \cos a x} a + \frac m a \int x^{m - 1} \cos a x \rd x + C       | c = from $(1)$ }} {{eqn | r = \frac {- x^m \cos a x} a + \frac m a \paren {\frac {x^{m - 1} \sin a x} a - \frac {m - 1} a \int x^{m - 2} \sin a x \rd x} + C       | c = from $(2)$ }} {{eqn | r = \frac {- x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \sin a x \rd x + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \sin a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = a \cos a x       | c = Derivative of $\sin a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac 1 {x^n}       | c =  }} {{eqn | r = x^{-n}       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^{-n + 1} } {- n + 1}       | c = Primitive of Power }} {{eqn | r = \frac {-1} {\paren {n - 1} x^{n - 1} }       | c = simplifying }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\sin a x} {x^n} \rd x       | r = \sin a x \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } - \int \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } \paren {a \cos a x} \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {-\sin a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cos a x} {x^{n - 1} } \rd x       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \frac 1 {\sin^{n - 2} a x}       | c =  }} {{eqn | r = \sin^{- n + 2} a x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = a \left({- n + 2}\right) \sin^{- n + 1} a x \cos a x       | c = Derivative of $\sin a x$, Derivative of Power, Chain Rule for Derivatives }} {{eqn | r = \frac {-a \left({n - 2}\right) \cos a x} {\sin^{n - 1} a x}       | c = simplifying }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = \frac 1 {\sin^2 a x}       | c =  }} {{eqn | r = \csc^2 a x       | c = Cosecant is $\dfrac 1 \sin$ }} {{eqn | ll= \implies       | l = v       | r = \frac {-\cot a x} a       | c = Primitive of $\csc^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\sin^n a x}       | r = \int \frac {\mathrm d x} {\sin^{n - 2} a x \sin^2 a x}       | c =  }} {{eqn | r = \left({\frac 1 {\sin^{n - 2} a x} }\right) \left({\frac {-\cot a x} a}\right) - \int \left({\frac {-\cot a x} a}\right) \left({\frac {-a \left({n - 2}\right) \cos a x} {\sin^{n - 1} a x} }\right) \ \mathrm d x       | c = Integration by Parts }} {{eqn | r = \frac {-\cos a x} {a \sin^{n - 1} a x} - \int \frac {\left({n - 2}\right) \cos^2 a x} {\sin^n a x} \ \mathrm d x       | c = Cotangent is $\dfrac {\cos}{\sin}$ }} {{eqn | r = \frac {-\cos a x} {a \sin^{n - 1} a x} - \int \frac {\left({n - 2}\right) \left({1 - \sin^2 a x}\right)} {\sin^n a x} \ \mathrm d x       | c = Sum of $\sin^2$ and $\cos^2$ }} {{eqn | r = \frac {-\cos a x} {a \sin^{n - 1} a x} - \left({n - 2}\right) \int \frac {\mathrm d x} {\sin^n a x} + \left({n - 2}\right) \int \frac {\mathrm d x} {\sin^{n - 2} a x}       | c = Linear Combination of Integrals }} {{eqn | ll= \implies       | l = \left({n - 1}\right) \int \frac {\mathrm d x} {\sin^n a x}       | r = \frac {-\cos a x} {a \sin^{n - 1} a x} + \left({n - 2}\right) \int \frac {\mathrm d x} {\sin^{n - 2} a x}       | c = gathering terms }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {\sin^n a x}       | r = \frac {- \cos a x} {a \left({n - 1}\right) \sin^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\mathrm d x} {\sin^{n - 2} a x}       | c = dividing by $n - 1$ }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 1       | c = Derivative of Identity Function }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \cos a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\sin a x} a       | c = Primitive of $\cos a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \map \cos {a x} \rd x       | r = x \paren {\frac {\sin a x} a} - \int \paren {\frac {\sin a x} a} \times 1 \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x \sin a x} a - \frac 1 a \int \sin a x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x \sin a x} a - \frac 1 a \paren {\frac {-\cos a x} a} + C       | c = Primitive of $\sin a x$ }} {{eqn | r = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C       | c = simplification }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^2       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 2 x       | c = Derivative of Power }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \cos a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\sin a x} a       | c = Primitive of $\cos a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^2 \map \cos {a x} \rd x       | r = x^2 \paren {\frac {\sin a x} a} - \int \paren {2 x \frac {\sin a x} a} \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2 \sin a x} a - \frac 2 a \int x \sin a x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x^2 \sin a x} a - \frac 2 a \paren {\frac {\sin a x} {a^2} - \frac {x \cos a x} a} + C       | c = Primitive of $x \sin a x$ }} {{eqn | r = \frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x + C       | c = simplification }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^3       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 3 x^2       | c = Derivative of Power }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \cos a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\sin a x} a       | c = Primitive of $\cos a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^3 \map \cos {a x} \rd x       | r = x^3 \paren {\frac {\sin a x} a} - \int \paren {3 x^2 \frac {\sin a x} a} \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^3 \sin a x} a - \frac 3 a \int x^2 \sin a x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x^3 \sin a x} a - \frac 3 a \paren {\frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x} + C       | c = Primitive of $x^2 \sin a x$ }} {{eqn | r = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C       | c = simplification }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\cos a x \rd x} x       | r = \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {a x}^{2 k} }{\paren {2 k}!} } \rd x       | c = Definition of Real Cosine Function }} {{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k a^{2 k} } {\paren {2 k}!} \int \frac 1 x \paren {x^{2 k} } \rd x       | c = Linear Combination of Integrals }} {{eqn | n = 1       | r = \int \frac {\rd x} x + \sum_{k \mathop = 1}^\infty \frac {\paren {-1}^k a^{2 k} } {\paren {2 k}!} \int \paren {x^{2 k - 1} } \rd x       | c = extracting case for $k = 0$ }} {{eqn | r = \int \frac {\d x} x + \sum_{k \mathop = 1}^\infty \frac {\paren {-1}^k a^{2 k} } {\paren {2 k}!} \frac {x^{2 k} } {2 k} + C       | c = Primitive of Power }} {{eqn | r = \int \frac {\d x} x + \sum_{k \mathop \ge 1} \frac {\paren {-1}^k \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C       | c = simplifying }} {{eqn | r = \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {-1}^k \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C       | c = Primitive of Reciprocal }} {{end-eqn}} The validity of $(1)$ follows from Cosine Function is Absolutely Convergent. {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \cos a x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = -a \sin a x       | c = Derivative of $\cos a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = \frac 1 {x^2}       | c =  }} {{eqn | ll= \implies       | l = v       | r = - \frac 1 x       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\cos a x \ \mathrm d x} {x^2}       | r = \int \cos a x \frac 1 {x^2} \ \mathrm d x       | c =  }} {{eqn | r = \cos a x \left({- \frac 1 x}\right) - \int \left({- \frac 1 x}\right) \left({-a \sin a x}\right) \ \mathrm d x       | c = Integration by Parts }} {{eqn | r = \frac {-\cos a x} x - a \int \frac {\sin a x \ \mathrm d x} x       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {\cos x}       | r = \int \sec x \rd x       | c = {{Defof|Real Secant Function}} }} {{eqn | r = \ln \size {\sec x + \tan x} + C       | c = Primitive of $\sec x$: Secant plus Tangent Form }} {{eqn | ll= \leadsto       | l = \int \frac {\d x} {\cos a x}       | r = \frac 1 a \ln \size {\sec a x + \tan a x} + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {\cos x}       | r = \int \sec x \rd x       | c = {{Defof|Real Secant Function}} }} {{eqn | r = \ln \size {\map \tan {\frac \pi 4 + \frac x 2} } + C       | c = Primitive of $\sec x$: Tangent plus Angle Form }} {{eqn | ll= \leadsto       | l = \int \frac {\d x} {\cos a x}       | r = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn 	| l = \int \frac {x \rd x}{\cos a x}  	| r = \int x \sec a x \rd x 	| c = {{Defof|Secant Function}} }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C 	| c = Primitive of $x \sec a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cos^2 x \, \mathrm d x       | r = \frac x 2 + \frac {\sin 2 x} 4 + C       | c = Primitive of $\cos^2 x$ }} {{eqn | ll= \implies       | l = \int \cos^2 a x \, \mathrm d x       | r = \frac 1 a \left({\frac {a x} 2 + \frac {\sin 2 a x} 4}\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac x 2 + \frac {\sin 2 a x} {4 a} + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u}{\mathrm d x}       | r = 1       | c = Derivative of Identity Function }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v}{\mathrm d x}       | r = \cos^2 a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac x 2 + \frac {\sin 2 a x} {4 a}       | c = Primitive of $\cos^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \cos^2 a x \ \mathrm d x       | r = x \left({\frac x 2 + \frac {\sin 2 a x} {4 a} }\right) - \int \left({\frac x 2 + \frac {\sin 2 a x} {4 a} }\right) \times 1 \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2} 2 + \frac {x \sin 2 a x} {4 a} - \frac 1 2 \int x \ \mathrm d x - \frac 1 {4 a} \int \sin 2 a x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x^2} 2 + \frac {x \sin 2 a x} {4 a} - \frac {x^2} 4 - \frac 1 {4 a} \int \sin 2 a x \ \mathrm d x + C       | c = Primitive of Power }} {{eqn | r = \frac {x^2} 2 + \frac {x \sin 2 a x} {4 a} - \frac {x^2} 4 - \frac 1 {4 a} \frac {\left({-\cos 2 a x}\right)} {2 a} + C       | c = Primitive of $\sin a x$ }} {{eqn | r = \frac {x^2} 4 + \frac {x \sin 2 a x} {4 a} + \frac {\cos 2 a x} {8 a^2} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cos^3 a x \rd x       | r = \int \left({\frac {3 \cos a x + \cos 3 a x} 4}\right) \rd x       | c = Power Reduction Formula for $\cos^3$ }} {{eqn | r = \frac 3 4 \int \cos a x \rd x + \frac 1 4 \int \cos 3 a x \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 3 4 \left({\frac {\sin a x} a}\right) + \frac 1 4 \left({\frac {\sin 3 a x} {3 a} }\right) + C       | c = Primitive of $\cos a x$ }} {{eqn | r = \frac {3 \sin a x} {4 a} + \frac 1 {12 a} \left({\sin 3 a x}\right) + C       | c = simplifying }} {{eqn | r = \frac {3 \sin a x} {4 a} + \frac 1 {12 a} \left({3 \sin a x - 4 \sin^3 a x}\right) + C       | c = Triple Angle Formula for Sine }} {{eqn | r = \frac {3 \sin a x} {4 a} + \frac {\sin a x} {4 a} - \frac {\sin^3 a x} {3 a} + C       | c = multipying out }} {{eqn | r = \frac {\sin a x} a - \frac {\sin^3 a x} {3 a} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sin^4 a x \ \mathrm d x       | r = \int \left({\frac {3 + 4 \cos 2 x + \cos 4 x} 8}\right) \ \mathrm d x       | c = Power Reduction Formula for $\cos^4$ }} {{eqn | r = \frac 3 8 \int \mathrm d x + \frac 1 2 \int \cos 2 a x \ \mathrm d x + \frac 1 8 \int \cos 4 a x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac {3 x} 8 + \frac 1 2 \int \cos 2 a x \ \mathrm d x + \frac 1 8 \int \cos 4 a x \ \mathrm d x + C       | c = Primitive of Constant }} {{eqn | r = \frac {3 x} 8 + \frac 1 2 \left({\frac {\sin 2 a x} {2 a} }\right) + \frac 1 8 \left({\frac {\sin 4 a x} {4 a} }\right) + C       | c = Primitive of $\cos a x$ }} {{eqn | r = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\cos^2 a x}       | r = \int \sec^2 a x \ \mathrm d x       | c = Definition of Cosecant }} {{eqn | r = \frac {\tan a x} a + C       | c = Primitive of $\sec^2 a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sec^2 x \ \mathrm d x       | r = \tan x + C       | c = Primitive of $\sec^2 x$ }} {{eqn | ll= \implies       | l = \int \sec^2 a x \ \mathrm d x       | r = \frac 1 a \left({\tan a x}\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\tan a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sec x \rd x       | r = \ln \size {\map \tan {\frac \pi 4 + \frac x 2} }       | c = Primitive of $\sec x$: Tangent plus Angle Form }} {{eqn | ll= \leadsto       | l = \int \sec a x \rd x       | r = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sec x \rd x       | r = \ln \size {\sec x + \tan x}       | c = Primitive of $\sec x$: Secant plus Tangent Form }} {{eqn | ll= \leadsto       | l = \int \sec a x \rd x       | r = \frac 1 a \ln \size {\sec a x + \tan a x} + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u}{\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \sec^{n - 2} a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = a \paren {n - 2} \sec^{n - 3} a x \sec a x \tan a x       | c = Derivative of Power, Derivative of $\sec$, Chain Rule for Derivatives }} {{eqn | r = a \paren {n - 2} \sec^{n - 2} a x \tan a x       | c =  }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \sec^2 a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\tan a x} a       | c = Primitive of $\sec^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \sec^n a x \rd x       | r = \int \sec^{n - 2} a x \sec^2 a x \rd x       | c =  }} {{eqn | r = \sec^{n - 2} a x \paren {\frac {\tan a x} a}       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \int \paren {\frac {\tan a x} a} \paren {a \paren {n - 2} \sec^{n - 2} a x \tan a x} \rd x       | c =  }} {{eqn | r = \frac {\sec^{n - 2} a x \tan a x} a - \paren {n - 2} \int \tan^2 a x \sec^{n - 2} a x \rd x       | c = simplifying }} {{eqn | r = \frac {\sec^{n - 2} a x \tan a x} a - \paren {n - 2} \int \paren {\sec^2 a x - 1} \sec^{n - 2} a x \rd x       | c = Difference of $\sec^2$ and $\tan^2$ }} {{eqn | r = \frac {\sec^{n - 2} a x \tan a x} a - \paren {n - 2} \int \sec^n a x \rd x       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= +       | r = \paren {n - 2} \int \sec^{n - 2} a x \rd x       | c =  }} {{eqn | ll= \leadsto       | l = \paren {n - 1} \int \sec^n a x \rd x       | r = \frac {\sec^{n - 2} a x \tan a x} a + \paren {n - 2} \int \sec^{n - 2} a x \rd x       | c = gathering terms }} {{eqn | l = \int \sec^n a x \rd x       | r = \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       | c = dividing by $n - 1$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {\cos^3 a x}       | r = \int \sec^3 a x \rd x       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \int \sec a x \rd x       | c = Primitive of $\sec^n a x$ }} {{eqn | r = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 2 \int \sec a x \rd x       | c = Secant is $\dfrac 1 \cos$ and Tangent is $\dfrac \sin \cos$ }} {{eqn | r = \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       | c = Primitive of $\sec a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cos a x \cos p x \rd x       | r = \int \paren {\frac {\map \cos {a x - p x} + \map \cos {a x + p x} } 2} \rd x       | c = Simpson's Formula for Cosine by Cosine }} {{eqn | r = \frac 1 2 \int \map \cos {\paren {a - p} x} \rd x + \frac 1 2 \int \map \cos {\paren {a + p} x} \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \frac {\map \sin {\paren {a - p} x} } {a - p} + \frac 1 2 \frac {\map \sin {\paren {a + p} x} } {a + p} + C       | c = Primitive of $\cos a x$ }} {{eqn | r = \frac {\map \sin {\paren {a - p} x} } {2 \paren {a - p} } + \frac {\map \sin {\paren {a + p} x} } {2 \paren {a + p} } + C       | c = tidying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = u       | r = \tan \frac x 2       | c =  }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {1 - \cos x}       | r = \int \frac {\dfrac {2 \ \mathrm d u} {1 + u^2} } {1 - \dfrac {1 - u^2} {1 + u^2} }       | c = Weierstrass Substitution }} {{eqn | r = \int \frac {2 \ \mathrm d u} {1 + u^2 - \left({1 - u^2}\right)}       | c = multiplying top and bottom by $1 + u^2$ }} {{eqn | r = \int \frac {\mathrm d u} {u^2}       | c = simplifying }} {{eqn | r = \frac {-1} u + C       | c = Primitive of Power }} {{eqn | r = \frac {-1} {\tan \dfrac x 2} + C       | c = substituting for $u$ }} {{eqn | r = -\cot \frac x 2 + C       | c = Cotangent is Reciprocal of Tangent }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {1 - \cos a x}       | r = \frac {-1} a \cot \frac {a x} 2 + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 1       | c = Primitive of Power }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac 1 {1 - \cos a x}       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {-1} a \cot \frac {a x} 2       | c = Primitive of $\dfrac 1 {1 - \cos a x}$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {x \rd x} {1 - \cos a x}       | r = \int u \rd v       | c =  }} {{eqn | r = x \paren {\frac {-1} a \cot \frac {a x} 2} - \int \paren {\frac {-1} a \cot \frac {a x} 2} \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {-x} a \cot \frac {a x} 2 - \frac {-1} a \paren {\frac 2 a \ln \size {\sin \frac {a x} 2} } + C       | c = Primitive of $\cot a x$ }} {{eqn | r = \frac {-x} a \cot \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sin \frac {a x} 2} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = u       | r = \tan \frac x 2       | c =  }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {1 + \cos x}       | r = \int \frac {\dfrac {2 \ \mathrm d u} {1 + u^2} } {1 + \dfrac {1 - u^2} {1 + u^2} }       | c = Weierstrass Substitution }} {{eqn | r = \int \frac {2 \ \mathrm d u} {1 + u^2 + \left({1 - u^2}\right)}       | c = multiplying top and bottom by $1 + u^2$ }} {{eqn | r = \int \mathrm d u       | c = simplifying }} {{eqn | r = u + C       | c = Primitive of Constant }} {{eqn | r = \tan \frac x 2 + C       | c = substituting for $u$ }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {1 + \cos a x}       | r = \frac 1 a \tan \frac {a x} 2 + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}