The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.
Generally, if the function sin x {\displaystyle \sin x} is any trigonometric function, and cos x {\displaystyle \cos x} is its derivative,
∫ a cos n x d x = a n sin n x + C {\displaystyle \int a\cos nx\,dx={\frac {a}{n}}\sin nx+C}
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
Integrands involving only sine
- ∫ sin a x d x = − 1 a cos a x + C {\displaystyle \int \sin ax\,dx=-{\frac {1}{a}}\cos ax+C}
- ∫ sin 2 a x d x = x 2 − 1 4 a sin 2 a x + C = x 2 − 1 2 a sin a x cos a x + C {\displaystyle \int \sin ^{2}{ax}\,dx={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}-{\frac {1}{2a}}\sin ax\cos ax+C}
- ∫ sin 3 a x d x = cos 3 a x 12 a − 3 cos a x 4 a + C {\displaystyle \int \sin ^{3}{ax}\,dx={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C}
- ∫ x sin 2 a x d x = x 2 4 − x 4 a sin 2 a x − 1 8 a 2 cos 2 a x + C {\displaystyle \int x\sin ^{2}{ax}\,dx={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C}
- ∫ x 2 sin 2 a x d x = x 3 6 − ( x 2 4 a − 1 8 a 3 ) sin 2 a x − x 4 a 2 cos 2 a x + C {\displaystyle \int x^{2}\sin ^{2}{ax}\,dx={\frac {x^{3}}{6}}-\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax-{\frac {x}{4a^{2}}}\cos 2ax+C}
- ∫ x sin a x d x = sin a x a 2 − x cos a x a + C {\displaystyle \int x\sin ax\,dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C}
- ∫ ( sin b 1 x ) ( sin b 2 x ) d x = sin ( ( b 2 − b 1 ) x ) 2 ( b 2 − b 1 ) − sin ( ( b 1 + b 2 ) x ) 2 ( b 1 + b 2 ) + C (for | b 1 | ≠ | b 2 | ) {\displaystyle \int (\sin b_{1}x)(\sin b_{2}x)\,dx={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(for }}|b_{1}|\neq |b_{2}|{\mbox{)}}}
- ∫ sin n a x d x = − sin n − 1 a x cos a x n a + n − 1 n ∫ sin n − 2 a x d x (for n > 0 ) {\displaystyle \int \sin ^{n}{ax}\,dx=-{\frac {\sin ^{n-1}ax\cos ax}{na}}+{\frac {n-1}{n}}\int \sin ^{n-2}ax\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}}
- ∫ d x sin a x = − 1 a ln | csc a x + cot a x | + C {\displaystyle \int {\frac {dx}{\sin ax}}=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}
- ∫ d x sin n a x = cos a x a ( 1 − n ) sin n − 1 a x + n − 2 n − 1 ∫ d x sin n − 2 a x (for n > 1 ) {\displaystyle \int {\frac {dx}{\sin ^{n}ax}}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}}
- ∫ x n sin a x d x = − x n a cos a x + n a ∫ x n − 1 cos a x d x = ∑ k = 0 2 k ≤ n ( − 1 ) k + 1 x n − 2 k a 1 + 2 k n ! ( n − 2 k ) ! cos a x + ∑ k = 0 2 k + 1 ≤ n ( − 1 ) k x n − 1 − 2 k a 2 + 2 k n ! ( n − 2 k − 1 ) ! sin a x = − ∑ k = 0 n x n − k a 1 + k n ! ( n − k ) ! cos ( a x + k π 2 ) (for n > 0 ) {\displaystyle {\begin{aligned}\int x^{n}\sin ax\,dx&=-{\frac {x^{n}}{a}}\cos ax+{\frac {n}{a}}\int x^{n-1}\cos ax\,dx\\&=\sum _{k=0}^{2k\leq n}(-1)^{k+1}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\cos ax+\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-1-2k}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\sin ax\\&=-\sum _{k=0}^{n}{\frac {x^{n-k}}{a^{1+k}}}{\frac {n!}{(n-k)!}}\cos \left(ax+k{\frac {\pi }{2}}\right)\qquad {\mbox{(for }}n>0{\mbox{)}}\end{aligned}}}
- ∫ sin a x x d x = ∑ n = 0 ∞ ( − 1 ) n ( a x ) 2 n + 1 ( 2 n + 1 ) ⋅ ( 2 n + 1 ) ! + C {\displaystyle \int {\frac {\sin ax}{x}}\,dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C}
- ∫ sin a x x n d x = − sin a x ( n − 1 ) x n − 1 + a n − 1 ∫ cos a x x n − 1 d x {\displaystyle \int {\frac {\sin ax}{x^{n}}}\,dx=-{\frac {\sin ax}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}\int {\frac {\cos ax}{x^{n-1}}}\,dx}
- ∫ sin ( a x 2 + b x + c ) d x = { a π 2 cos ( b 2 − 4 a c 4 a ) S ( 2 a x + b 2 a π ) + a π 2 sin ( b 2 − 4 a c 4 a ) C ( 2 a x + b 2 a π ) t o b 2 − 4 a c > 0 a π 2 cos ( b 2 − 4 a c 4 a ) S ( 2 a x + b 2 a π ) − a π 2 sin ( b 2 − 4 a c 4 a ) C ( 2 a x + b 2 a π ) t o b 2 − 4 a c < 0 f o r a ╱ = 0 , a > 0 {\displaystyle \int {\sin {\mathrm {(} }{ax}^{2}\mathrm {+} {bx}\mathrm {+} {c}{\mathrm {)} }{dx}}\mathrm {=} \left\{{\begin{aligned}&{{\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\cos \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){S}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\mathrm {+} {\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\sin \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){C}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {>} }\;{0}}\\&{{\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\cos \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){S}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\mathrm {-} {\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\sin \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){C}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {<} }\;{0}}\end{aligned}}\right.\;\;{for}\;{a}\diagup \!\!\!\!{\mathrm {=} }{0}{\mathrm {,} }\;{a}{\mathrm {>} }{0}}
- ∫ d x 1 ± sin a x = 1 a tan ( a x 2 ∓ π 4 ) + C {\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}
- ∫ x d x 1 + sin a x = x a tan ( a x 2 − π 4 ) + 2 a 2 ln | cos ( a x 2 − π 4 ) | + C {\displaystyle \int {\frac {x\,dx}{1+\sin ax}}={\frac {x}{a}}\tan \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{a^{2}}}\ln \left|\cos \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)\right|+C}
- ∫ x d x 1 − sin a x = x a cot ( π 4 − a x 2 ) + 2 a 2 ln | sin ( π 4 − a x 2 ) | + C {\displaystyle \int {\frac {x\,dx}{1-\sin ax}}={\frac {x}{a}}\cot \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)+{\frac {2}{a^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)\right|+C}
- ∫ sin a x d x 1 ± sin a x = ± x + 1 a tan ( π 4 ∓ a x 2 ) + C {\displaystyle \int {\frac {\sin ax\,dx}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}
Integrands involving only cosine
- ∫ cos a x d x = 1 a sin a x + C {\displaystyle \int \cos ax\,dx={\frac {1}{a}}\sin ax+C}
- ∫ cos 2 a x d x = x 2 + 1 4 a sin 2 a x + C = x 2 + 1 2 a sin a x cos a x + C {\displaystyle \int \cos ^{2}{ax}\,dx={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}+{\frac {1}{2a}}\sin ax\cos ax+C}
- ∫ cos n a x d x = cos n − 1 a x sin a x n a + n − 1 n ∫ cos n − 2 a x d x (for n > 0 ) {\displaystyle \int \cos ^{n}ax\,dx={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}}
- ∫ x cos a x d x = cos a x a 2 + x sin a x a + C {\displaystyle \int x\cos ax\,dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C}
- ∫ x 2 cos 2 a x d x = x 3 6 + ( x 2 4 a − 1 8 a 3 ) sin 2 a x + x 4 a 2 cos 2 a x + C {\displaystyle \int x^{2}\cos ^{2}{ax}\,dx={\frac {x^{3}}{6}}+\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax+{\frac {x}{4a^{2}}}\cos 2ax+C}
- ∫ x n cos a x d x = x n sin a x a − n a ∫ x n − 1 sin a x d x = ∑ k = 0 2 k + 1 ≤ n ( − 1 ) k x n − 2 k − 1 a 2 + 2 k n ! ( n − 2 k − 1 ) ! cos a x + ∑ k = 0 2 k ≤ n ( − 1 ) k x n − 2 k a 1 + 2 k n ! ( n − 2 k ) ! sin a x = ∑ k = 0 n ( − 1 ) ⌊ k / 2 ⌋ x n − k a 1 + k n ! ( n − k ) ! cos ( a x − ( − 1 ) k + 1 2 π 2 ) = ∑ k = 0 n x n − k a 1 + k n ! ( n − k ) ! sin ( a x + k π 2 ) (for n > 0 ) {\displaystyle {\begin{aligned}\int x^{n}\cos ax\,dx&={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\,dx\\&=\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-2k-1}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\cos ax+\sum _{k=0}^{2k\leq n}(-1)^{k}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\sin ax\\&=\sum _{k=0}^{n}(-1)^{\lfloor k/2\rfloor }{\frac {x^{n-k}}{a^{1+k}}}{\frac {n!}{(n-k)!}}\cos \left(ax-{\frac {(-1)^{k}+1}{2}}{\frac {\pi }{2}}\right)\\&=\sum _{k=0}^{n}{\frac {x^{n-k}}{a^{1+k}}}{\frac {n!}{(n-k)!}}\sin \left(ax+k{\frac {\pi }{2}}\right)\qquad {\mbox{(for }}n>0{\mbox{)}}\end{aligned}}}
- ∫ cos a x x d x = ln | a x | + ∑ k = 1 ∞ ( − 1 ) k ( a x ) 2 k 2 k ⋅ ( 2 k ) ! + C {\displaystyle \int {\frac {\cos ax}{x}}\,dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C}
- ∫ cos a x x n d x = − cos a x ( n − 1 ) x n − 1 − a n − 1 ∫ sin a x x n − 1 d x (for n ≠ 1 ) {\displaystyle \int {\frac {\cos ax}{x^{n}}}\,dx=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ d x cos a x = 1 a ln | tan ( a x 2 + π 4 ) | + C {\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
- ∫ d x cos n a x = sin a x a ( n − 1 ) cos n − 1 a x + n − 2 n − 1 ∫ d x cos n − 2 a x (for n > 1 ) {\displaystyle \int {\frac {dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}}
- ∫ d x 1 + cos a x = 1 a tan a x 2 + C {\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C}
- ∫ d x 1 − cos a x = − 1 a cot a x 2 + C {\displaystyle \int {\frac {dx}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}
- ∫ x d x 1 + cos a x = x a tan a x 2 + 2 a 2 ln | cos a x 2 | + C {\displaystyle \int {\frac {x\,dx}{1+\cos ax}}={\frac {x}{a}}\tan {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\cos {\frac {ax}{2}}\right|+C}
- ∫ x d x 1 − cos a x = − x a cot a x 2 + 2 a 2 ln | sin a x 2 | + C {\displaystyle \int {\frac {x\,dx}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C}
- ∫ cos a x d x 1 + cos a x = x − 1 a tan a x 2 + C {\displaystyle \int {\frac {\cos ax\,dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C}
- ∫ cos a x d x 1 − cos a x = − x − 1 a cot a x 2 + C {\displaystyle \int {\frac {\cos ax\,dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}
- ∫ ( cos a 1 x ) ( cos a 2 x ) d x = sin ( ( a 2 − a 1 ) x ) 2 ( a 2 − a 1 ) + sin ( ( a 2 + a 1 ) x ) 2 ( a 2 + a 1 ) + C (for | a 1 | ≠ | a 2 | ) {\displaystyle \int (\cos a_{1}x)(\cos a_{2}x)\,dx={\frac {\sin((a_{2}-a_{1})x)}{2(a_{2}-a_{1})}}+{\frac {\sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}}
Integrands involving only tangent
- ∫ tan a x d x = − 1 a ln | cos a x | + C = 1 a ln | sec a x | + C {\displaystyle \int \tan ax\,dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C}
- ∫ tan 2 x d x = tan x − x + C {\displaystyle \int \tan ^{2}{x}\,dx=\tan {x}-x+C}
- ∫ tan n a x d x = 1 a ( n − 1 ) tan n − 1 a x − ∫ tan n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \tan ^{n}ax\,dx={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ d x q tan a x + p = 1 p 2 + q 2 ( p x + q a ln | q sin a x + p cos a x | ) + C (for p 2 + q 2 ≠ 0 ) {\displaystyle \int {\frac {dx}{q\tan ax+p}}={\frac {1}{p^{2}+q^{2}}}(px+{\frac {q}{a}}\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(for }}p^{2}+q^{2}\neq 0{\mbox{)}}}
- ∫ d x tan a x ± 1 = ± x 2 + 1 2 a ln | sin a x ± cos a x | + C {\displaystyle \int {\frac {dx}{\tan ax\pm 1}}=\pm {\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax\pm \cos ax|+C}
- ∫ tan a x d x tan a x ± 1 = x 2 ∓ 1 2 a ln | sin a x ± cos a x | + C {\displaystyle \int {\frac {\tan ax\,dx}{\tan ax\pm 1}}={\frac {x}{2}}\mp {\frac {1}{2a}}\ln |\sin ax\pm \cos ax|+C}
Integrands involving only secant
Further information: Integral of the secant function
- ∫ sec a x d x = 1 a ln | sec a x + tan a x | + C = 1 a ln | tan ( a x 2 + π 4 ) | + C = 1 a artanh ( sin a x ) + C {\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C={\frac {1}{a}}\ln {\left|\tan {\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)}\right|}+C={\frac {1}{a}}\operatorname {artanh} {\left(\sin {ax}\right)}+C}
- ∫ sec 2 x d x = tan x + C {\displaystyle \int \sec ^{2}{x}\,dx=\tan {x}+C}
- ∫ sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C . {\displaystyle \int \sec ^{3}{x}\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C.}
- ∫ sec n a x d x = sec n − 2 a x tan a x a ( n − 1 ) + n − 2 n − 1 ∫ sec n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-2}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}}
- ∫ d x sec x + 1 = x − tan x 2 + C {\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}
- ∫ d x sec x − 1 = − x − cot x 2 + C {\displaystyle \int {\frac {dx}{\sec {x}-1}}=-x-\cot {\frac {x}{2}}+C}
Integrands involving only cosecant
- ∫ csc a x d x = − 1 a ln | csc a x + cot a x | + C = 1 a ln | csc a x − cot a x | + C = 1 a ln | tan ( a x 2 ) | + C {\displaystyle \int \csc {ax}\,dx=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C={\frac {1}{a}}\ln {\left|\csc {ax}-\cot {ax}\right|}+C={\frac {1}{a}}\ln {\left|\tan {\left({\frac {ax}{2}}\right)}\right|}+C}
- ∫ csc 2 x d x = − cot x + C {\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C}
- ∫ csc 3 x d x = − 1 2 csc x cot x − 1 2 ln | csc x + cot x | + C = − 1 2 csc x cot x + 1 2 ln | csc x − cot x | + C {\displaystyle \int \csc ^{3}{x}\,dx=-{\frac {1}{2}}\csc x\cot x-{\frac {1}{2}}\ln |\csc x+\cot x|+C=-{\frac {1}{2}}\csc x\cot x+{\frac {1}{2}}\ln |\csc x-\cot x|+C}
- ∫ csc n a x d x = − csc n − 2 a x cot a x a ( n − 1 ) + n − 2 n − 1 ∫ csc n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-2}{ax}\cot {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}}
- ∫ d x csc x + 1 = x − 2 cot x 2 + 1 + C {\displaystyle \int {\frac {dx}{\csc {x}+1}}=x-{\frac {2}{\cot {\frac {x}{2}}+1}}+C}
- ∫ d x csc x − 1 = − x + 2 cot x 2 − 1 + C {\displaystyle \int {\frac {dx}{\csc {x}-1}}=-x+{\frac {2}{\cot {\frac {x}{2}}-1}}+C}
Integrands involving only cotangent
- ∫ cot a x d x = 1 a ln | sin a x | + C {\displaystyle \int \cot ax\,dx={\frac {1}{a}}\ln |\sin ax|+C}
- ∫ cot 2 x d x = − cot x − x + C {\displaystyle \int \cot ^{2}{x}\,dx=-\cot {x}-x+C}
- ∫ cot n a x d x = − 1 a ( n − 1 ) cot n − 1 a x − ∫ cot n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \cot ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ d x 1 + cot a x = ∫ tan a x d x tan a x + 1 = x 2 − 1 2 a ln | sin a x + cos a x | + C {\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\,dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C}
- ∫ d x 1 − cot a x = ∫ tan a x d x tan a x − 1 = x 2 + 1 2 a ln | sin a x − cos a x | + C {\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\,dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C}
Integrands involving both sine and cosine
An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules.
- ∫ d x cos a x ± sin a x = 1 a 2 ln | tan ( a x 2 ± π 8 ) | + C {\displaystyle \int {\frac {dx}{\cos ax\pm \sin ax}}={\frac {1}{a{\sqrt {2}}}}\ln \left|\tan \left({\frac {ax}{2}}\pm {\frac {\pi }{8}}\right)\right|+C}
- ∫ d x ( cos a x ± sin a x ) 2 = 1 2 a tan ( a x ∓ π 4 ) + C {\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C}
- ∫ d x ( cos x + sin x ) n = 1 2 ( n − 1 ) ( sin x − cos x ( cos x + sin x ) n − 1 + ( n − 2 ) ∫ d x ( cos x + sin x ) n − 2 ) {\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}}}={\frac {1}{2(n-1)}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}+(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2}}}\right)}
- ∫ cos a x d x cos a x + sin a x = x 2 + 1 2 a ln | sin a x + cos a x | + C {\displaystyle \int {\frac {\cos ax\,dx}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
- ∫ cos a x d x cos a x − sin a x = x 2 − 1 2 a ln | sin a x − cos a x | + C {\displaystyle \int {\frac {\cos ax\,dx}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}
- ∫ sin a x d x cos a x + sin a x = x 2 − 1 2 a ln | sin a x + cos a x | + C {\displaystyle \int {\frac {\sin ax\,dx}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
- ∫ sin a x d x cos a x − sin a x = − x 2 − 1 2 a ln | sin a x − cos a x | + C {\displaystyle \int {\frac {\sin ax\,dx}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}
- ∫ cos a x d x ( sin a x ) ( 1 + cos a x ) = − 1 4 a tan 2 a x 2 + 1 2 a ln | tan a x 2 | + C {\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}
- ∫ cos a x d x ( sin a x ) ( 1 − cos a x ) = − 1 4 a cot 2 a x 2 − 1 2 a ln | tan a x 2 | + C {\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}
- ∫ sin a x d x ( cos a x ) ( 1 + sin a x ) = 1 4 a cot 2 ( a x 2 + π 4 ) + 1 2 a ln | tan ( a x 2 + π 4 ) | + C {\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1+\sin ax)}}={\frac {1}{4a}}\cot ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
- ∫ sin a x d x ( cos a x ) ( 1 − sin a x ) = 1 4 a tan 2 ( a x 2 + π 4 ) − 1 2 a ln | tan ( a x 2 + π 4 ) | + C {\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1-\sin ax)}}={\frac {1}{4a}}\tan ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
- ∫ ( sin a x ) ( cos a x ) d x = 1 2 a sin 2 a x + C {\displaystyle \int (\sin ax)(\cos ax)\,dx={\frac {1}{2a}}\sin ^{2}ax+C}
- ∫ ( sin a 1 x ) ( cos a 2 x ) d x = − cos ( ( a 1 − a 2 ) x ) 2 ( a 1 − a 2 ) − cos ( ( a 1 + a 2 ) x ) 2 ( a 1 + a 2 ) + C (for | a 1 | ≠ | a 2 | ) {\displaystyle \int (\sin a_{1}x)(\cos a_{2}x)\,dx=-{\frac {\cos((a_{1}-a_{2})x)}{2(a_{1}-a_{2})}}-{\frac {\cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}}
- ∫ ( sin n a x ) ( cos a x ) d x = 1 a ( n + 1 ) sin n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int (\sin ^{n}ax)(\cos ax)\,dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
- ∫ ( sin a x ) ( cos n a x ) d x = − 1 a ( n + 1 ) cos n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int (\sin ax)(\cos ^{n}ax)\,dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
- ∫ ( sin n a x ) ( cos m a x ) d x = − ( sin n − 1 a x ) ( cos m + 1 a x ) a ( n + m ) + n − 1 n + m ∫ ( sin n − 2 a x ) ( cos m a x ) d x (for m , n > 0 ) = ( sin n + 1 a x ) ( cos m − 1 a x ) a ( n + m ) + m − 1 n + m ∫ ( sin n a x ) ( cos m − 2 a x ) d x (for m , n > 0 ) {\displaystyle {\begin{aligned}\int (\sin ^{n}ax)(\cos ^{m}ax)\,dx&=-{\frac {(\sin ^{n-1}ax)(\cos ^{m+1}ax)}{a(n+m)}}+{\frac {n-1}{n+m}}\int (\sin ^{n-2}ax)(\cos ^{m}ax)\,dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\\&={\frac {(\sin ^{n+1}ax)(\cos ^{m-1}ax)}{a(n+m)}}+{\frac {m-1}{n+m}}\int (\sin ^{n}ax)(\cos ^{m-2}ax)\,dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\end{aligned}}}
- ∫ d x ( sin a x ) ( cos a x ) = 1 a ln | tan a x | + C {\displaystyle \int {\frac {dx}{(\sin ax)(\cos ax)}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}
- ∫ d x ( sin a x ) ( cos n a x ) = 1 a ( n − 1 ) cos n − 1 a x + ∫ d x ( sin a x ) ( cos n − 2 a x ) (for n ≠ 1 ) {\displaystyle \int {\frac {dx}{(\sin ax)(\cos ^{n}ax)}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+\int {\frac {dx}{(\sin ax)(\cos ^{n-2}ax)}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ d x ( sin n a x ) ( cos a x ) = − 1 a ( n − 1 ) sin n − 1 a x + ∫ d x ( sin n − 2 a x ) ( cos a x ) (for n ≠ 1 ) {\displaystyle \int {\frac {dx}{(\sin ^{n}ax)(\cos ax)}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+\int {\frac {dx}{(\sin ^{n-2}ax)(\cos ax)}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ sin a x d x cos n a x = 1 a ( n − 1 ) cos n − 1 a x + C (for n ≠ 1 ) {\displaystyle \int {\frac {\sin ax\,dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ sin 2 a x d x cos a x = − 1 a sin a x + 1 a ln | tan ( π 4 + a x 2 ) | + C {\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}
- ∫ sin 2 a x d x cos n a x = sin a x a ( n − 1 ) cos n − 1 a x − 1 n − 1 ∫ d x cos n − 2 a x (for n ≠ 1 ) {\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ sin 2 x 1 + cos 2 x d x = 2 arctangant ( tan x 2 ) − x (for x in ] − π 2 ; + π 2 [ ) = 2 arctangant ( tan x 2 ) − arctangant ( tan x ) (this time x being any real number ) {\displaystyle {\begin{aligned}\int {\frac {\sin ^{2}x}{1+\cos ^{2}x}}\,dx&={\sqrt {2}}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}}}\right)-x\qquad {\mbox{(for x in}}]-{\frac {\pi }{2}};+{\frac {\pi }{2}}[{\mbox{)}}\\&={\sqrt {2}}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}}}\right)-\operatorname {arctangant} \left(\tan x\right)\qquad {\mbox{(this time x being any real number }}{\mbox{)}}\end{aligned}}}
- ∫ sin n a x d x cos a x = − sin n − 1 a x a ( n − 1 ) + ∫ sin n − 2 a x d x cos a x (for n ≠ 1 ) {\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\,dx}{\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ sin n a x d x cos m a x = { sin n + 1 a x a ( m − 1 ) cos m − 1 a x − n − m + 2 m − 1 ∫ sin n a x d x cos m − 2 a x (for m ≠ 1 ) sin n − 1 a x a ( m − 1 ) cos m − 1 a x − n − 1 m − 1 ∫ sin n − 2 a x d x cos m − 2 a x (for m ≠ 1 ) − sin n − 1 a x a ( n − m ) cos m − 1 a x + n − 1 n − m ∫ sin n − 2 a x d x cos m a x (for m ≠ n ) {\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ^{m}ax}}={\begin{cases}{\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}ax\,dx}{\cos ^{m-2}ax}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\{\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m-2}ax}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m}ax}}&{\mbox{(for }}m\neq n{\mbox{)}}\end{cases}}}
- ∫ cos a x d x sin n a x = − 1 a ( n − 1 ) sin n − 1 a x + C (for n ≠ 1 ) {\displaystyle \int {\frac {\cos ax\,dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ cos 2 a x d x sin a x = 1 a ( cos a x + ln | tan a x 2 | ) + C {\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C}
- ∫ cos 2 a x d x sin n a x = − 1 n − 1 ( cos a x a sin n − 1 a x + ∫ d x sin n − 2 a x ) (for n ≠ 1 ) {\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax}}+\int {\frac {dx}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
- ∫ cos n a x d x sin m a x = { − cos n + 1 a x a ( m − 1 ) sin m − 1 a x − n − m + 2 m − 1 ∫ cos n a x d x sin m − 2 a x (for n ≠ 1 ) − cos n − 1 a x a ( m − 1 ) sin m − 1 a x − n − 1 m − 1 ∫ cos n − 2 a x d x sin m − 2 a x (for m ≠ 1 ) cos n − 1 a x a ( n − m ) sin m − 1 a x + n − 1 n − m ∫ cos n − 2 a x d x sin m a x (for m ≠ n ) {\displaystyle \int {\frac {\cos ^{n}ax\,dx}{\sin ^{m}ax}}={\begin{cases}-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\cos ^{n}ax\,dx}{\sin ^{m-2}ax}}&{\mbox{(for }}n\neq 1{\mbox{)}}\\-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m-2}ax}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\{\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m}ax}}&{\mbox{(for }}m\neq n{\mbox{)}}\end{cases}}}
Integrands involving both sine and tangent
- ∫ ( sin a x ) ( tan a x ) d x = 1 a ( ln | sec a x + tan a x | − sin a x ) + C {\displaystyle \int (\sin ax)(\tan ax)\,dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C}
- ∫ tan n a x d x sin 2 a x = 1 a ( n − 1 ) tan n − 1 ( a x ) + C (for n ≠ 1 ) {\displaystyle \int {\frac {\tan ^{n}ax\,dx}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
Integrand involving both cosine and tangent
- ∫ tan n a x d x cos 2 a x = 1 a ( n + 1 ) tan n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int {\frac {\tan ^{n}ax\,dx}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
Integrand involving both sine and cotangent
- ∫ cot n a x d x sin 2 a x = − 1 a ( n + 1 ) cot n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int {\frac {\cot ^{n}ax\,dx}{\sin ^{2}ax}}=-{\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
Integrand involving both cosine and cotangent
- ∫ cot n a x d x cos 2 a x = 1 a ( 1 − n ) tan 1 − n a x + C (for n ≠ 1 ) {\displaystyle \int {\frac {\cot ^{n}ax\,dx}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
Integrand involving both secant and tangent
- ∫ ( sec x ) ( tan x ) d x = sec x + C {\displaystyle \int (\sec x)(\tan x)\,dx=\sec x+C}
Integrand involving both cosecant and cotangent
- ∫ ( csc x ) ( cot x ) d x = − csc x + C {\displaystyle \int (\csc x)(\cot x)\,dx=-\csc x+C}
Integrals in a quarter period
Using the beta function B ( a , b ) {\displaystyle B(a,b)} one can write
- ∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = 1 2 B ( n + 1 2 , 1 2 ) = { n − 1 n ⋅ n − 3 n − 2 ⋯ 3 4 ⋅ 1 2 ⋅ π 2 , if n is even n − 1 n ⋅ n − 3 n − 2 ⋯ 4 5 ⋅ 2 3 , if n is odd and more than 1 1 , if n = 1 {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {1}{2}}B\left({\frac {n+1}{2}},{\frac {1}{2}}\right)={\begin{cases}{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {3}{4}}\cdot {\frac {1}{2}}\cdot {\frac {\pi }{2}},&{\text{if }}n{\text{ is even}}\\{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {4}{5}}\cdot {\frac {2}{3}},&{\text{if }}n{\text{ is odd and more than 1}}\\1,&{\text{if }}n=1\end{cases}}}
Using the modified Struve functions L α ( x ) {\displaystyle L_{\alpha }(x)} and modified Bessel functions I α ( x ) {\displaystyle I_{\alpha }(x)} one can write
- ∫ 0 π 2 exp ( k ⋅ sin ( x ) ) d x = π 2 ( I 0 ( k ) + L 0 ( k ) ) {\displaystyle \int _{0}^{\frac {\pi }{2}}\exp(k\cdot \sin(x))\,dx={\frac {\pi }{2}}\left(I_{0}(k)+L_{0}(k)\right)}
Integrals with symmetric limits
- ∫ − c c sin x d x = 0 {\displaystyle \int _{-c}^{c}\sin {x}\,dx=0}
- ∫ − c c cos x d x = 2 ∫ 0 c cos x d x = 2 ∫ − c 0 cos x d x = 2 sin c {\displaystyle \int _{-c}^{c}\cos {x}\,dx=2\int _{0}^{c}\cos {x}\,dx=2\int _{-c}^{0}\cos {x}\,dx=2\sin {c}}
- ∫ − c c tan x d x = 0 {\displaystyle \int _{-c}^{c}\tan {x}\,dx=0}
- ∫ − a 2 a 2 x 2 cos 2 n π x a d x = a 3 ( n 2 π 2 − 6 ) 24 n 2 π 2 (for n = 1 , 3 , 5... ) {\displaystyle \int _{-{\frac {a}{2}}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=1,3,5...{\mbox{)}}}
- ∫ − a 2 a 2 x 2 sin 2 n π x a d x = a 3 ( n 2 π 2 − 6 ( − 1 ) n ) 24 n 2 π 2 = a 3 24 ( 1 − 6 ( − 1 ) n n 2 π 2 ) (for n = 1 , 2 , 3 , . . . ) {\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6(-1)^{n})}{24n^{2}\pi ^{2}}}={\frac {a^{3}}{24}}(1-6{\frac {(-1)^{n}}{n^{2}\pi ^{2}}})\qquad {\mbox{(for }}n=1,2,3,...{\mbox{)}}}
Integral over a full circle
- ∫ 0 2 π sin 2 m + 1 x cos n x d x = 0 n , m ∈ Z {\displaystyle \int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{n}{x}\,dx=0\!\qquad n,m\in \mathbb {Z} }
- ∫ 0 2 π sin m x cos 2 n + 1 x d x = 0 n , m ∈ Z {\displaystyle \int _{0}^{2\pi }\sin ^{m}{x}\cos ^{2n+1}{x}\,dx=0\!\qquad n,m\in \mathbb {Z} }
See also
References
Bresock, Krista (2022-01-01). "Student Understanding of the Definite Integral When Solving Calculus Volume Problems". Graduate Theses, Dissertations, and Problem Reports. doi:10.33915/etd.11491. https://researchrepository.wvu.edu/etd/11491 ↩