三角関数の原始関数の一覧
∫ 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}\;\mathrm {d} x={\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 ( | b 1 | ≠ | b 2 | ) {\displaystyle \int \sin b_{1}x\sin b_{2}x\;\mathrm {d} x={\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{(}}|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 ( n > 2 ) {\displaystyle \int \sin ^{n}{ax}\;\mathrm {d} x=-{\frac {\sin ^{n-1}ax\cos ax}{na}}+{\frac {n-1}{n}}\int \sin ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(}}n>2{\mbox{)}}\,\!}
∫ d x cos  a x ± sin  a x = 1 a 2 ln  | tan  ( a x 2 ± π 8 ) | + C {\displaystyle \int {\frac {\mathrm {d} x}{\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 {\mathrm {d} x}{(\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 {\mathrm {d} x}{(\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 {\mathrm {d} x}{(\cos x+\sin x)^{n-2}}}\right)}
∫ d x 1 + cos  a x = 1 a tan  a x 2 + C {\displaystyle \int {\frac {\mathrm {d} x}{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 {\mathrm {d} x}{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\;\mathrm {d} x}{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\;\mathrm {d} x}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C}