三角関数の原始関数の一覧
∫ sin  a x d x = − 1 a cos  a x + C {\displaystyle \int \sin ax\;\mathrm {d} x=-{\frac {1}{a}}\cos ax+C\,\!}
∫ sin 2  a x d x = x 2 − 1 4 a sin  2 a x + C = x 2 − 1 2 a sin  a x cos  a x + C {\displaystyle \int \sin ^{2}{ax}\;\mathrm {d} x={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}-{\frac {1}{2a}}\sin ax\cos ax+C\!}
∫ sin 3  a x d x = cos  3 a x 12 a − 3 cos  a x 4 a + C {\displaystyle \int \sin ^{3}{ax}\;\mathrm {d} x={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C\!}
∫ x sin 2  a x d x = x 2 4 − x 4 a sin  2 a x − 1 8 a 2 cos  2 a x + C {\displaystyle \int x\sin ^{2}{ax}\;\mathrm {d} x={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C\!}
∫ 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}
∫ csc 2  x d x = − cot  x + C {\displaystyle \int \csc ^{2}{x}\,\mathrm {d} x=-\cot {x}+C}
∫ csc n  a x d x = − csc n − 1  a x cos  a x a ( n − 1 ) + n − 2 n − 1 ∫ csc n − 2  a x d x ( n ≠ 1 ) {\displaystyle \int \csc ^{n}{ax}\,\mathrm {d} x=-{\frac {\csc ^{n-1}{ax}\cos {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,\mathrm {d} x\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ cos  a x d x cos  a x + sin  a x = x 2 + 1 2 a ln  | sin  a x + cos  a x | + C {\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
∫ cos  a x d x cos  a x − sin  a x = x 2 − 1 2 a ln  | sin  a x − cos  a x | + C {\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}
∫ sin  a x d x cos  a x + sin  a x = x 2 − 1 2 a ln  | sin  a x + cos  a x | + C {\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
∫ d x sin n  a x cos  a x = − 1 a ( n − 1 ) sin n − 1  a x + ∫ d x sin n − 2  a x cos  a x ( n ≠ 1 ) {\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{n}ax\cos ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax\cos ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ sin  a x d x cos n  a x = 1 a ( n − 1 ) cos n − 1  a x + C ( n ≠ 1 ) {\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ sin 2  a x d x cos  a x = − 1 a sin  a x + 1 a ln  | tan  ( π 4 + a x 2 ) | + C {\displaystyle \int {\frac {\sin ^{2}ax\;\mathrm {d} x}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}