三角関数の原始関数の一覧
∫ sin  a x tan  a x d x = 1 a ( ln  | sec  a x + tan  a x | − sin  a x ) + C {\displaystyle \int \sin ax\tan ax\;\mathrm {d} x={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}
∫ tan n  a x d x sin 2  a x = 1 a ( n − 1 ) tan n − 1  ( a x ) + C ( n ≠ 1 ) {\displaystyle \int {\frac {\tan ^{n}ax\;\mathrm {d} x}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ cot  a x d x = 1 a ln  | sin  a x | + C {\displaystyle \int \cot ax\;\mathrm {d} x={\frac {1}{a}}\ln \left|\sin ax\right|+C\,\!}
∫ sec  a x d x = 1 a ln  | sec  a x + tan  a x | + C = 1 a gd − 1  ( a x ) + C gd − 1  x {\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\quad \operatorname {gd} ^{-1}x} :グーデルマン関数の逆関数
∫ csc  a x d x = − 1 a ln  | csc  a x + cot  a x | + C {\displaystyle \int \csc {ax}\,\mathrm {d} x=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}
∫ sin n  a x d x cos m  a x = sin n + 1  a x a ( m − 1 ) cos m − 1  a x − n − m + 2 m − 1 ∫ sin n  a x d x cos m − 2  a x ( m ≠ 1 ) {\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m}ax}}={\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m-2}ax}}\qquad {\mbox{(}}m\neq 1{\mbox{)}}\,\!}
または ∫ sin n  a x d x cos m  a x = − sin n − 1  a x a ( n − m ) cos m − 1  a x + n − 1 n − m ∫ sin n − 2  a x d x cos m  a x ( m ≠ n ) {\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m}ax}}=-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}ax\;\mathrm {d} x}{\cos ^{m}ax}}\qquad {\mbox{(}}m\neq n{\mbox{)}}\,\!}
∫ d x cos  a x = 1 a ln  | tan  ( a x 2 + π 4 ) | + C = 1 a gd − 1  ( a x ) + C {\displaystyle \int {\frac {\mathrm {d} x}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C={\frac {1}{a}}\operatorname {gd} ^{-1}(ax)+C}
∫ d x cos n  a x = sin  a x a ( n − 1 ) cos n − 1  a x + n − 2 n − 1 ∫ d x cos n − 2  a x ( n > 1 ) {\displaystyle \int {\frac {\mathrm {d} x}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}ax}}\qquad {\mbox{(}}n>1{\mbox{)}}\,\!}
∫ sin 2  a x d x cos n  a x = sin  a x a ( n − 1 ) cos n − 1  a x − 1 n − 1 ∫ d x cos n − 2  a x ( n ≠ 1 ) {\displaystyle \int {\frac {\sin ^{2}ax\;\mathrm {d} x}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ sin n  a x d x cos  a x = − sin n − 1  a x a ( n − 1 ) + ∫ sin n − 2  a x d x cos  a x ( n ≠ 1 ) {\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\;\mathrm {d} x}{\cos ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}