三角関数の原始関数の一覧
∫ 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\;\mathrm {d} x}{\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\;\mathrm {d} x}{\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 d x = − 1 a cos  ( a x ) + C {\displaystyle \int \sin ax\;\mathrm {d} x=-{\frac {1}{a}}\cos \left(ax\right)+C}
∫ cos  a x d x = 1 a sin  a x + C {\displaystyle \int \cos ax\;\mathrm {d} x={\frac {1}{a}}\sin ax+C}
∫ tan  a x d x = − 1 a ln  | cos  ( a x ) | + C = 1 a ln  | sec  ( a x ) | + C {\displaystyle \int \tan ax\;\mathrm {d} x=-{\frac {1}{a}}\ln \left|\cos \left(ax\right)\right|+C={\frac {1}{a}}\ln \left|\sec \left(ax\right)\right|+C\,\!}