三角関数の原始関数の一覧
∫ cot n  a x d x cos 2  a x = 1 a ( 1 − n ) tan 1 − n  a x + C ( n ≠ 1 ) {\displaystyle \int {\frac {\cot ^{n}ax\;\mathrm {d} x}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ cot n  a x d x sin 2  a x = − 1 a ( n + 1 ) cot n + 1  a x + C ( n ≠ − 1 ) {\displaystyle \int {\frac {\cot ^{n}ax\;\mathrm {d} x}{\sin ^{2}ax}}=-{\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(}}n\neq -1{\mbox{)}}\,\!}
∫ 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 ) ( n ≠ 1 ) {\displaystyle \int {\frac {\cos ^{2}ax\;\mathrm {d} x}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax)}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(}}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 ( 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-m-2}{m-1}}\int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\sin ^{m-2}ax}}\qquad {\mbox{(}}m\neq 1{\mbox{)}}\,\!}
∫ sec  a x d x = 1 a ln  | sec  a x + tan  a x | + C = 1 a gd − 1  ( a x ) + C {\displaystyle \int \sec {ax}\,\mathrm {d} x={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C={\frac {1}{a}}\operatorname {gd} ^{-1}(ax)+C}
∫ sec 2  x d x = tan  x + C {\displaystyle \int \sec ^{2}{x}\,\mathrm {d} x=\tan {x}+C}
∫ sec n  a x d x = sec n − 2  a x tan  a x a ( n − 1 ) + n − 2 n − 1 ∫ sec n − 2  a x d x ( n ≠ 1 ) {\displaystyle \int \sec ^{n}{ax}\,\mathrm {d} x={\frac {\sec ^{n-2}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,\mathrm {d} x\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}