三角関数の原始関数の一覧
∫ cos  a x d x 1 + cos  a x = x − 1 a tan  a x 2 + C {\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}
∫ cos  a x d x 1 − cos  a x = − x − 1 a cot  a x 2 + C {\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}
∫ cos  a 1 x cos  a 2 x d x = sin  ( a 1 − a 2 ) x 2 ( a 1 − a 2 ) + sin  ( a 1 + a 2 ) x 2 ( a 1 + a 2 ) + C ( | a 1 | ≠ | a 2 | ) {\displaystyle \int \cos a_{1}x\cos a_{2}x\;\mathrm {d} x={\frac {\sin(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+{\frac {\sin(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(}}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}
∫ cot  a x d x = 1 a ln  | sin  a x | + C {\displaystyle \int \cot ax\;\mathrm {d} x={\frac {1}{a}}\ln |\sin ax|+C\,\!}
∫ cot n  a x d x = − 1 a ( n − 1 ) cot n − 1  a x − ∫ cot n − 2  a x d x ( n ≠ 1 ) {\displaystyle \int \cot ^{n}ax\;\mathrm {d} x=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ d x 1 + cot  a x = ∫ tan  a x d x tan  a x + 1 {\displaystyle \int {\frac {\mathrm {d} x}{1+\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax+1}}\,\!}
∫ d x 1 − cot  a x = ∫ tan  a x d x tan  a x − 1 {\displaystyle \int {\frac {\mathrm {d} x}{1-\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax-1}}\,\!}
または ∫ sin n  a x d x cos m  a x = sin n − 1  a x a ( m − 1 ) cos m − 1  a x − n − 1 m − 1 ∫ sin n − 2  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-1}{m-1}}\int {\frac {\sin ^{n-2}ax\;\mathrm {d} x}{\cos ^{m-2}ax}}\qquad {\mbox{(}}m\neq 1{\mbox{)}}\,\!}
∫ cos  a x d x sin n  a x = − 1 a ( n − 1 ) sin n − 1  a x + C ( n ≠ 1 ) {\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}
∫ cos 2  a x d x sin  a x = 1 a ( cos  a x + ln  | tan  a x 2 | ) + C {\displaystyle \int {\frac {\cos ^{2}ax\;\mathrm {d} x}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C}
∫ − c c sin  x d x = 0 {\displaystyle \int _{-c}^{c}\sin {x}\;\mathrm {d} x=0\!}
∫ − c c cos  x d x = 2 ∫ 0 c cos  x d x = 2 ∫ − c 0 cos  x d x = 2 sin  c {\displaystyle \int _{-c}^{c}\cos {x}\;\mathrm {d} x=2\int _{0}^{c}\cos {x}\;\mathrm {d} x=2\int _{-c}^{0}\cos {x}\;\mathrm {d} x=2\sin {c}\!}
∫ − c c tan  x d x = 0 {\displaystyle \int _{-c}^{c}\tan {x}\;\mathrm {d} x=0\!}
∫ − a 2 a 2 x 2 cos 2  n π x a d x = a 3 ( n 2 π 2 − 6 ) 24 n 2 π 2 ( n = 1 , 3 , 5... ) {\displaystyle \int _{-{\frac {a}{2}}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\;\mathrm {d} x={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(}}n=1,3,5...{\mbox{)}}\,\!}