قائمة تكاملات الدوال المثلثية

 تكاملات مثلثية تحتوي فقط على الجيب (جا)

${\displaystyle{\displaystyle\int\sin ax\;dx=-{\frac{1}{a}}\cos ax+C\,\!}}$

${\displaystyle{\displaystyle\int\sin^{2}{ax}\;dx={\frac{x}{2}}-{\frac{1}{4a}}\sin 2ax+C={\frac{x}{2}}-{\frac{1}{2a}}\sin ax\cos ax+C\!}}$

${\displaystyle{\displaystyle\int x\sin^{2}{ax}\;dx={\frac{x^{2}}{4}}-{\frac{x}{4a}}\sin 2ax-{\frac{1}{8a^{2}}}\cos 2ax+C\!}}$

${\displaystyle{\displaystyle\int x^{2}\sin^{2}{ax}\;dx={\frac{x^{3}}{6}}-\left({\frac{x^{2}}{4a}}-{\frac{1}{8a^{3}}}\right)\sin 2ax-{\frac{x}{4a^{2}}}\cos 2ax+C\!}}$

${\displaystyle{\displaystyle\int\sin b_{1}x\sin b_{2}x\;dx={\frac{\sin((b_{1}-b_{2})x)}{2(b_{1}-b_{2})}}-{\frac{\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad{\mbox{(for }}|b_{1}|\neq|b_{2}|{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int\sin^{n}{ax}\;dx=-{\frac{\sin^{n-1}ax\cos ax}{na}}+{\frac{n-1}{n}}\int\sin^{n-2}ax\;dx\qquad{\mbox{(for }}n>0{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sin ax}}={\frac{1}{a}}\ln\left|\tan{\frac{ax}{2}}\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sin^{n}ax}}={\frac{\cos ax}{a(1-n)\sin^{n-1}ax}}+{\frac{n-2}{n-1}}\int{\frac{dx}{\sin^{n-2}ax}}\qquad{\mbox{(for }}n>1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int x\sin ax\;dx={\frac{\sin ax}{a^{2}}}-{\frac{x\cos ax}{a}}+C\,\!}}$

${\displaystyle{\displaystyle\int x^{n}\sin ax\;dx=-{\frac{x^{n}}{a}}\cos ax+{\frac{n}{a}}\int x^{n-1}\cos ax\;dx\qquad{\mbox{(for }}n>0{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int_{\frac{-a}{2}}^{\frac{a}{2}}x^{2}\sin^{2}{\frac{n\pi x}{a}}\;dx={\frac{a^{3}(n^{2}\pi^{2}-6)}{24n^{2}\pi^{2}}}\qquad{\mbox{(for }}n=2,4,6...{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax}{x}}dx=\sum_{n=0}^{\infty}(-1)^{n}{\frac{(ax)^{2n+1}}{(2n+1)\cdot(2n+1)!}}+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax}{x^{n}}}dx=-{\frac{\sin ax}{(n-1)x^{n-1}}}+{\frac{a}{n-1}}\int{\frac{\cos ax}{x^{n-1}}}dx\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{1\pm\sin ax}}={\frac{1}{a}}\tan\left({\frac{ax}{2}}\mp{\frac{\pi}{4}}\right)+C}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{1+\sin ax}}={\frac{x}{a}}\tan\left({\frac{ax}{2}}-{\frac{\pi}{4}}\right)+{\frac{2}{a^{2}}}\ln\left|\cos\left({\frac{ax}{2}}-{\frac{\pi}{4}}\right)\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{1-\sin ax}}={\frac{x}{a}}\cot\left({\frac{\pi}{4}}-{\frac{ax}{2}}\right)+{\frac{2}{a^{2}}}\ln\left|\sin\left({\frac{\pi}{4}}-{\frac{ax}{2}}\right)\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax\;dx}{1\pm\sin ax}}=\pm x+{\frac{1}{a}}\tan\left({\frac{\pi}{4}}\mp{\frac{ax}{2}}\right)+C}}$

 Antiderivatives containing only cosine

${\displaystyle{\displaystyle\int\cos ax\;dx={\frac{1}{a}}\sin ax+C\,\!}}$

${\displaystyle{\displaystyle\int\cos^{n}ax\;dx={\frac{\cos^{n-1}ax\sin ax}{na}}+{\frac{n-1}{n}}\int\cos^{n-2}ax\;dx\qquad{\mbox{(for }}n>0{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int x\cos ax\;dx={\frac{\cos ax}{a^{2}}}+{\frac{x\sin ax}{a}}+C\,\!}}$

${\displaystyle{\displaystyle\int\cos^{2}{ax}\;dx={\frac{x}{2}}+{\frac{1}{4a}}\sin 2ax+C={\frac{x}{2}}+{\frac{1}{2a}}\sin ax\cos ax+C\!}}$

${\displaystyle{\displaystyle\int x^{2}\cos^{2}{ax}\;dx={\frac{x^{3}}{6}}+\left({\frac{x^{2}}{4a}}-{\frac{1}{8a^{3}}}\right)\sin 2ax+{\frac{x}{4a^{2}}}\cos 2ax+C\!}}$

${\displaystyle{\displaystyle\int x^{n}\cos ax\;dx={\frac{x^{n}\sin ax}{a}}-{\frac{n}{a}}\int x^{n-1}\sin ax\;dx\,\!}}$

${\displaystyle{\displaystyle\int_{\frac{-a}{2}}^{\frac{a}{2}}x^{2}\cos^{2}{\frac{n\pi x}{a}}\;dx={\frac{a^{3}(n^{2}\pi^{2}-6)}{24n^{2}\pi^{2}}}\qquad{\mbox{(for }}n=1,3,5...{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax}{x}}dx=\ln|ax|+\sum_{k=1}^{\infty}(-1)^{k}{\frac{(ax)^{2k}}{2k\cdot(2k)!}}+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax}{x^{n}}}dx=-{\frac{\cos ax}{(n-1)x^{n-1}}}-{\frac{a}{n-1}}\int{\frac{\sin ax}{x^{n-1}}}dx\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\cos ax}}={\frac{1}{a}}\ln\left|\tan\left({\frac{ax}{2}}+{\frac{\pi}{4}}\right)\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\cos^{n}ax}}={\frac{\sin ax}{a(n-1)\cos^{n-1}ax}}+{\frac{n-2}{n-1}}\int{\frac{dx}{\cos^{n-2}ax}}\qquad{\mbox{(for }}n>1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{1+\cos ax}}={\frac{1}{a}}\tan{\frac{ax}{2}}+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{1-\cos ax}}=-{\frac{1}{a}}\cot{\frac{ax}{2}}+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{1+\cos ax}}={\frac{x}{a}}\tan{\frac{ax}{2}}+{\frac{2}{a^{2}}}\ln\left|\cos{\frac{ax}{2}}\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{1-\cos ax}}=-{\frac{x}{a}}\cot{\frac{ax}{2}}+{\frac{2}{a^{2}}}\ln\left|\sin{\frac{ax}{2}}\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax\;dx}{1+\cos ax}}=x-{\frac{1}{a}}\tan{\frac{ax}{2}}+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax\;dx}{1-\cos ax}}=-x-{\frac{1}{a}}\cot{\frac{ax}{2}}+C\,\!}}$

${\displaystyle{\displaystyle\int\cos a_{1}x\cos a_{2}x\;dx={\frac{\sin(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+{\frac{\sin(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}+C\qquad{\mbox{(for }}|a_{1}|\neq|a_{2}|{\mbox{)}}\,\!}}$

 Antiderivatives containing only tangent

${\displaystyle{\displaystyle\int\tan ax\;dx=-{\frac{1}{a}}\ln|\cos ax|+C={\frac{1}{a}}\ln|\sec ax|+C\,\!}}$

${\displaystyle{\displaystyle\int\tan^{n}ax\;dx={\frac{1}{a(n-1)}}\tan^{n-1}ax-\int\tan^{n-2}ax\;dx\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{q\tan ax+p}}={\frac{1}{p^{2}+q^{2}}}(px+{\frac{q}{a}}\ln|q\sin ax+p\cos ax|)+C\qquad{\mbox{(for }}p^{2}+q^{2}\neq 0{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\tan ax}}={\frac{1}{a}}\ln|\sin ax|+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\tan ax+1}}={\frac{x}{2}}+{\frac{1}{2a}}\ln|\sin ax+\cos ax|+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\tan ax-1}}=-{\frac{x}{2}}+{\frac{1}{2a}}\ln|\sin ax-\cos ax|+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\tan ax\;dx}{\tan ax+1}}={\frac{x}{2}}-{\frac{1}{2a}}\ln|\sin ax+\cos ax|+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\tan ax\;dx}{\tan ax-1}}={\frac{x}{2}}+{\frac{1}{2a}}\ln|\sin ax-\cos ax|+C\,\!}}$

 Antiderivatives containing only secant

${\displaystyle{\displaystyle\int\sec{ax}\,dx={\frac{1}{a}}\ln{\left|\sec{ax}+\tan{ax}\right|}+C}}$

${\displaystyle{\displaystyle\int\sec^{n}{ax}\,dx={\frac{\sec^{n-1}{ax}\sin{ax}}{a(n-1)}}\,+\,{\frac{n-2}{n-1}}\int\sec^{n-2}{ax}\,dx\qquad{\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int\sec^{n}{x}\,dx={\frac{\sec^{n-2}{x}\tan{x}}{n-1}}\,+\,{\frac{n-2}{n-1}}\int\sec^{n-2}{x}\,dx}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sec{x}+1}}=x-\tan{\frac{x}{2}}+C}}$

 Antiderivatives containing only cosecant

${\displaystyle{\displaystyle\int\csc{ax}\,dx={\frac{1}{a}}\ln{\left|\csc{ax}-\cot{ax}\right|}+C}}$

${\displaystyle{\displaystyle\int\csc^{2}{x}\,dx=-\cot{x}+C}}$

${\displaystyle{\displaystyle\int\csc^{n}{ax}\,dx=-{\frac{\csc^{n-1}{ax}\cos{ax}}{a(n-1)}}\,+\,{\frac{n-2}{n-1}}\int\csc^{n-2}{ax}\,dx\qquad{\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}}$

 Antiderivatives containing only cotangent

${\displaystyle{\displaystyle\int\cot ax\;dx={\frac{1}{a}}\ln|\sin ax|+C\,\!}}$

${\displaystyle{\displaystyle\int\cot^{n}ax\;dx=-{\frac{1}{a(n-1)}}\cot^{n-1}ax-\int\cot^{n-2}ax\;dx\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{1+\cot ax}}=\int{\frac{\tan ax\;dx}{\tan ax+1}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{1-\cot ax}}=\int{\frac{\tan ax\;dx}{\tan ax-1}}\,\!}}$

 Antiderivatives containing both sine and cosine

${\displaystyle{\displaystyle\int{\frac{dx}{\cos ax\pm\sin ax}}={\frac{1}{a{\sqrt{2}}}}\ln\left|\tan\left({\frac{ax}{2}}\pm{\frac{\pi}{8}}\right)\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{dx}{(\cos ax\pm\sin ax)^{2}}}={\frac{1}{2a}}\tan\left(ax\mp{\frac{\pi}{4}}\right)+C}}$

${\displaystyle{\displaystyle\int{\frac{dx}{(\cos x+\sin x)^{n}}}={\frac{1}{n-1}}\left({\frac{\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}-2(n-2)\int{\frac{dx}{(\cos x+\sin x)^{n-2}}}\right)}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax\;dx}{\cos ax+\sin ax}}={\frac{x}{2}}+{\frac{1}{2a}}\ln\left|\sin ax+\cos ax\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax\;dx}{\cos ax-\sin ax}}={\frac{x}{2}}-{\frac{1}{2a}}\ln\left|\sin ax-\cos ax\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax\;dx}{\cos ax+\sin ax}}={\frac{x}{2}}-{\frac{1}{2a}}\ln\left|\sin ax+\cos ax\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax\;dx}{\cos ax-\sin ax}}=-{\frac{x}{2}}-{\frac{1}{2a}}\ln\left|\sin ax-\cos ax\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax\;dx}{\sin ax(1+\cos ax)}}=-{\frac{1}{4a}}\tan^{2}{\frac{ax}{2}}+{\frac{1}{2a}}\ln\left|\tan{\frac{ax}{2}}\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax\;dx}{\sin ax(1+-\cos ax)}}=-{\frac{1}{4a}}\cot^{2}{\frac{ax}{2}}-{\frac{1}{2a}}\ln\left|\tan{\frac{ax}{2}}\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax\;dx}{\cos ax(1+\sin ax)}}={\frac{1}{4a}}\cot^{2}\left({\frac{ax}{2}}+{\frac{\pi}{4}}\right)+{\frac{1}{2a}}\ln\left|\tan\left({\frac{ax}{2}}+{\frac{\pi}{4}}\right)\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax\;dx}{\cos ax(1-\sin ax)}}={\frac{1}{4a}}\tan^{2}\left({\frac{ax}{2}}+{\frac{\pi}{4}}\right)-{\frac{1}{2a}}\ln\left|\tan\left({\frac{ax}{2}}+{\frac{\pi}{4}}\right)\right|+C}}$

${\displaystyle{\displaystyle\int\sin ax\cos ax\;dx={\frac{1}{2a}}\sin^{2}ax+C\,\!}}$

${\displaystyle{\displaystyle\int\sin a_{1}x\cos a_{2}x\;dx=-{\frac{\cos(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}-{\frac{\cos(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+C\qquad{\mbox{(for }}|a_{1}|\neq|a_{2}|{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int\sin^{n}ax\cos ax\;dx={\frac{1}{a(n+1)}}\sin^{n+1}ax+C\qquad{\mbox{(for }}n\neq-1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int\sin ax\cos^{n}ax\;dx=-{\frac{1}{a(n+1)}}\cos^{n+1}ax+C\qquad{\mbox{(for }}n\neq-1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int\sin^{n}ax\cos^{m}ax\;dx=-{\frac{\sin^{n-1}ax\cos^{m+1}ax}{a(n+m)}}+{\frac{n-1}{n+m}}\int\sin^{n-2}ax\cos^{m}ax\;dx\qquad{\mbox{(for }}m,n>0{\mbox{)}}\,\!}}$

also: ${\displaystyle{\displaystyle\int\sin^{n}ax\cos^{m}ax\;dx={\frac{\sin^{n+1}ax\cos^{m-1}ax}{a(n+m)}}+{\frac{m-1}{n+m}}\int\sin^{n}ax\cos^{m-2}ax\;dx\qquad{\mbox{(for }}m,n>0{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sin ax\cos ax}}={\frac{1}{a}}\ln\left|\tan ax\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sin ax\cos^{n}ax}}={\frac{1}{a(n-1)\cos^{n-1}ax}}+\int{\frac{dx}{\sin ax\cos^{n-2}ax}}\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sin^{n}ax\cos ax}}=-{\frac{1}{a(n-1)\sin^{n-1}ax}}+\int{\frac{dx}{\sin^{n-2}ax\cos ax}}\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\sin ax\;dx}{\cos^{n}ax}}={\frac{1}{a(n-1)\cos^{n-1}ax}}+C\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\sin^{2}ax\;dx}{\cos ax}}=-{\frac{1}{a}}\sin ax+{\frac{1}{a}}\ln\left|\tan\left({\frac{\pi}{4}}+{\frac{ax}{2}}\right)\right|+C}}$

${\displaystyle{\displaystyle\int{\frac{\sin^{2}ax\;dx}{\cos^{n}ax}}={\frac{\sin ax}{a(n-1)\cos^{n-1}ax}}-{\frac{1}{n-1}}\int{\frac{dx}{\cos^{n-2}ax}}\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\sin^{n}ax\;dx}{\cos ax}}=-{\frac{\sin^{n-1}ax}{a(n-1)}}+\int{\frac{\sin^{n-2}ax\;dx}{\cos ax}}\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\sin^{n}ax\;dx}{\cos^{m}ax}}={\frac{\sin^{n+1}ax}{a(m-1)\cos^{m-1}ax}}-{\frac{n-m+2}{m-1}}\int{\frac{\sin^{n}ax\;dx}{\cos^{m-2}ax}}\qquad{\mbox{(for }}m\neq 1{\mbox{)}}\,\!}}$

also: ${\displaystyle{\displaystyle\int{\frac{\sin^{n}ax\;dx}{\cos^{m}ax}}=-{\frac{\sin^{n-1}ax}{a(n-m)\cos^{m-1}ax}}+{\frac{n-1}{n-m}}\int{\frac{\sin^{n-2}ax\;dx}{\cos^{m}ax}}\qquad{\mbox{(for }}m\neq n{\mbox{)}}\,\!}}$

also: ${\displaystyle{\displaystyle\int{\frac{\sin^{n}ax\;dx}{\cos^{m}ax}}={\frac{\sin^{n-1}ax}{a(m-1)\cos^{m-1}ax}}-{\frac{n-1}{m-1}}\int{\frac{\sin^{n-2}ax\;dx}{\cos^{m-2}ax}}\qquad{\mbox{(for }}m\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\cos ax\;dx}{\sin^{n}ax}}=-{\frac{1}{a(n-1)\sin^{n-1}ax}}+C\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\cos^{2}ax\;dx}{\sin ax}}={\frac{1}{a}}\left(\cos ax+\ln\left|\tan{\frac{ax}{2}}\right|\right)+C}}$

${\displaystyle{\displaystyle\int{\frac{\cos^{2}ax\;dx}{\sin^{n}ax}}=-{\frac{1}{n-1}}\left({\frac{\cos ax}{a\sin^{n-1}ax)}}+\int{\frac{dx}{\sin^{n-2}ax}}\right)\qquad{\mbox{(for }}n\neq 1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{\cos^{n}ax\;dx}{\sin^{m}ax}}=-{\frac{\cos^{n+1}ax}{a(m-1)\sin^{m-1}ax}}-{\frac{n-m-2}{m-1}}\int{\frac{\cos^{n}ax\;dx}{\sin^{m-2}ax}}\qquad{\mbox{(for }}m\neq 1{\mbox{)}}\,\!}}$

also: ${\displaystyle{\displaystyle\int{\frac{\cos^{n}ax\;dx}{\sin^{m}ax}}={\frac{\cos^{n-1}ax}{a(n-m)\sin^{m-1}ax}}+{\frac{n-1}{n-m}}\int{\frac{\cos^{n-2}ax\;dx}{\sin^{m}ax}}\qquad{\mbox{(for }}m\neq n{\mbox{)}}\,\!}}$

also: ${\displaystyle{\displaystyle\int{\frac{\cos^{n}ax\;dx}{\sin^{m}ax}}=-{\frac{\cos^{n-1}ax}{a(m-1)\sin^{m-1}ax}}-{\frac{n-1}{m-1}}\int{\frac{\cos^{n-2}ax\;dx}{\sin^{m-2}ax}}\qquad{\mbox{(for }}m\neq 1{\mbox{)}}\,\!}}$

 Antiderivatives containing both sine and tangent

${\displaystyle{\displaystyle\int\sin ax\tan ax\;dx={\frac{1}{a}}(\ln|\sec ax+\tan ax|-\sin ax)+C\,\!}}$

${\displaystyle{\displaystyle\int{\frac{\tan^{n}ax\;dx}{\sin^{2}ax}}={\frac{1}{a(n-1)}}\tan^{n-1}(ax)+C\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

 Antiderivatives containing both cosine and tangent

${\displaystyle{\displaystyle\int{\frac{\tan^{n}ax\;dx}{\cos^{2}ax}}={\frac{1}{a(n+1)}}\tan^{n+1}ax+C\qquad{\mbox{(for }}n\neq-1{\mbox{)}}\,\!}}$

 Antiderivatives containing both sine and cotangent

${\displaystyle{\displaystyle\int{\frac{\cot^{n}ax\;dx}{\sin^{2}ax}}={\frac{1}{a(n+1)}}\cot^{n+1}ax+C\qquad{\mbox{(for }}n\neq-1{\mbox{)}}\,\!}}$

 Antiderivatives containing both cosine and cotangent

${\displaystyle{\displaystyle\int{\frac{\cot^{n}ax\;dx}{\cos^{2}ax}}={\frac{1}{a(1-n)}}\tan^{1-n}ax+C\qquad{\mbox{(for }}n\neq 1{\mbox{)}}\,\!}}$

 Antiderivatives with symmetric limits

${\displaystyle{\displaystyle\int_{-c}^{c}\sin{x}\;dx=0\!}}$

${\displaystyle{\displaystyle\int_{-c}^{c}\cos{x}\;dx=2\int_{0}^{c}\cos{x}\;dx=2\int_{-c}^{0}\cos{x}\;dx=2\sin{c}\!}}$

${\displaystyle{\displaystyle\int_{-c}^{c}\tan{x}\;dx=0\!}}$