三角関数の原始関数の一覧
または ∫ 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 ( m , n > 0 ) {\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;\mathrm {d} x={\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\;\mathrm {d} x\qquad {\mbox{(}}m,n>0{\mbox{)}}\,\!}
∫ d x sin  a x cos  a x = 1 a ln  | tan  a x | + C {\displaystyle \int {\frac {\mathrm {d} x}{\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 ( n ≠ 1 ) {\displaystyle \int {\frac {\mathrm {d} x}{\sin ax\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+\int {\frac {\mathrm {d} x}{\sin ax\cos ^{n-2}ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
}{(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}}\mathrm {d} x=\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 ( n ≠ 1 ) {\displaystyle \int {\frac {\cos ax}{x^{n}}}\mathrm {d} x=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}\mathrm {d} x\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ sin  a x cos  a x d x = − 1 2 a cos 2  a x + C {\displaystyle \int \sin ax\cos ax\;\mathrm {d} x=-{\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 ( | a 1 | ≠ | a 2 | ) {\displaystyle \int \sin a_{1}x\cos a_{2}x\;\mathrm {d} x=-{\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{(}}|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 ( n ≠ − 1 ) {\displaystyle \int \sin ^{n}ax\cos ax\;\mathrm {d} x={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(}}n\neq -1{\mbox{)}}\,\!}
∫ 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\;\mathrm {d} x}{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\;\mathrm {d} x}{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\;\mathrm {d} x}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}