Det følgende er en liste over integraler (antideriverte funksjoner ) av trigonometriske funksjoner . For antideriverte som involverer både eksponentialfunksjoner og trigonometriske funksjoner, se Liste over integraler av eksponentialfunksjoner. For en liste over antideriverte funksjoner, se lister over integraler. Se også trigonometrisk integral.
Generelt, hvis funksjonen sin ( x ) {\displaystyle \sin(x)} cos ( x ) {\displaystyle \cos(x)}
∫ a cos n x d x = a n sin n x + c {\displaystyle \int a\cos nx\;dx={\frac {a}{n}}\sin nx+c} I alle formler antas konstanten a å være forskjellige fra null, og C betegner integrasjonskonstanten.
Integrander som bare involverer sinus ∫ 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\!}
∫ 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\!}
∫ sin b 1 x sin b 2 x d x = sin ( ( b 1 − b 2 ) x ) 2 ( b 1 − b 2 ) − 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_{1}-b_{2})x)}{2(b_{1}-b_{2})}}-{\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 | tan a x 2 | + C {\displaystyle \int {\frac {dx}{\sin ax}}={\frac {1}{a}}\ln \left|\tan {\frac {ax}{2}}\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 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\,\!}
∫ 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 (for n > 0 ) {\displaystyle \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\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}
∫ − a 2 a 2 x 2 sin 2 n π x a d x = a 3 ( n 2 π 2 − 6 ) 24 n 2 π 2 (for n = 2 , 4 , 6... ) {\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)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=2,4,6...{\mbox{)}}\,\!}
∫ 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\,\!}
∫ 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} Integrander som bare involverer cosinus ∫ 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 {\displaystyle \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\!} ∫ 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 1 − a 2 ) x 2 ( a 1 − a 2 ) + sin ( a 1 + a 2 ) x 2 ( a 1 + a 2 ) + C (for | a 1 | ≠ | a 2 | ) {\displaystyle \int \cos a_{1}x\cos a_{2}x\;dx={\frac {\sin(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+{\frac {\sin(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!} Integrander som bare involverer tangens Integrander som bare involverer secans I det 17. århundre var integralet av secansfunksjonen temaet for en velkjent formodning fremsatt i 1640-årene av Henry Bond. Problemet ble løst av Isaac Barrow [1] kartografi . Se Integralet av secansfunksjonen.
∫ sec a x d x = 1 a ln | sec a x + tan a x | + C {\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C} ∫ sec 2 x d x = tan x + C {\displaystyle \int \sec ^{2}{x}\,dx=\tan {x}+C} ∫ sec n a x d x = sec n − 1 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-1}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!} ∫ sec n x d x = sec n − 2 x tan x n − 1 + n − 2 n − 1 ∫ sec n − 2 x d x {\displaystyle \int \sec ^{n}{x}\,dx={\frac {\sec ^{n-2}{x}\tan {x}}{n-1}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{x}\,dx} [2] ∫ 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} Integrander som bare involverer cosecans ∫ csc a x d x = − 1 a ln | csc a x + cot a x | + C {\displaystyle \int \csc {ax}\,dx=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C} ∫ csc 2 x d x = − cot x + C {\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C} ∫ csc n a x d x = − csc n − 1 a x cos 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-1}{ax}\cos {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 sin x 2 cos x 2 + sin x 2 + C {\displaystyle \int {\frac {dx}{\csc {x}+1}}=x-{\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}}+C} ∫ d x csc x − 1 = 2 sin x 2 cos x 2 − sin x 2 − x + C {\displaystyle \int {\frac {dx}{\csc {x}-1}}={\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}-x+C} ∫ 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 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 {\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax+1}}\,\!} ∫ d x 1 − cot a x = ∫ tan a x d x tan a x − 1 {\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax-1}}\,\!} ∫ 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 n − 1 ( sin x − cos x ( cos x + sin x ) n − 1 − 2 ( n − 2 ) ∫ d x ( cos x + sin x ) n − 2 ) {\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}}}={\frac {1}{n-1}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}-2(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 cos 2 a x + C {\displaystyle \int \sin ax\cos ax\;dx=-{\frac {1}{2a}}\cos ^{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 ) {\displaystyle \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{)}}\,\!} også: ∫ sin n a x cos m a x d x = 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 \int \sin ^{n}ax\cos ^{m}ax\;dx={\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{)}}\,\!} ∫ 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 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 ) {\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}={\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}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!} også: ∫ sin n a x d x cos m a x = − 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}}=-{\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}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!} også: ∫ sin n a x d x cos m a x = 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 ) {\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}={\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}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!} ∫ 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 m ≠ 1 ) {\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}=-{\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}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!} også: ∫ cos n a x d x sin m a x = 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}}={\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}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!} også: ∫ cos n a x d x sin m a x = − 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 ) {\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}=-{\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}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!} ∫ 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{)}}\,\!} ∫ 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{)}}\,\!} ∫ 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{)}}\,\!} ∫ 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{)}}\,\!}
Integraler med symmetriske grenser ∫ − 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{)}}\,\!}
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