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