قائمة تكاملات الدوال المثلثية

هذه قائمة ببعض تكاملات الدوال المثلثية. في كل هذه الصيغ نعتبر

$a$

غير منعدم و

$C$

هي ثابتة التكامل.

تكاملات مثلثية تحتوي فقط على الجيب

$\int \sin axdx=-\frac{1}{a}\cos ax+C$

$\int {\sin }^{2}axdx=\frac{x}{2}-\frac{1}{4a}\sin 2ax+C=\frac{x}{2}-\frac{1}{2a}\sin ax\cos ax+C$

$\int \sin a_{1}x\sin a_{2}xdx=\frac{\sin [(a_1-a_2)x]}{2(a_1-a_2)}-\frac{\sin [(a_1+a_2)x]}{2(a_1+a_2)}+C\text{(for }\left|a_1\right|\ne \left|a_2\right|\text{)}$

$\int {\sin }^{n}axdx=-\frac{{\sin }^{n-1}ax\cos ax}{na}+\frac{n-1}{n}\int {\sin }^{n-2}axdx\text{(for }n>0\text{)}$

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

$\int \frac{dx}{{\sin }^{n}ax}=\frac{\cos ax}{a(1-n){\sin }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\sin }^{n-2}ax}\text{(for }n>1\text{)}$

$\int x\sin axdx=\frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C$

$\int x^n\sin axdx=-\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos axdx\text{(for }n>0\text{)}$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\sin }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(for }n=2,4,6...\text{)}$

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

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

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

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

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

$\int \frac{\sin axdx}{1\pm \sin ax}=\pm x+\frac{1}{a}\tan (\frac{\pi }{4}\mp \frac{ax}{2})+C$

تكاملات مثلثية تحتوي فقط على جيب التمام

$\int \cos ax\mathrm{d}x=\frac{1}{a}\sin ax+C$

$\int {\cos }^{2}ax\mathrm{d}x=\frac{x}{2}+\frac{1}{4a}\sin 2ax+C=\frac{x}{2}+\frac{1}{2a}\sin ax\cos ax+C$

$\int {\cos }^{n}ax\mathrm{d}x=\frac{{\cos }^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int {\cos }^{n-2}ax\mathrm{d}x\text{(for }n>0\text{)}$

$\int x\cos ax\mathrm{d}x=\frac{\cos ax}{a^2}+\frac{x\sin ax}{a}+C$

$\int x^2{\cos }^{2}ax\mathrm{d}x=\frac{x^3}{6}+(\frac{x^2}{4a}-\frac{1}{8a^3})\sin 2ax+\frac{x}{4a^2}\cos 2ax+C$

$\int x^n\cos ax\mathrm{d}x=\frac{x^n\sin ax}{a}-\frac{n}{a}\int x^{n-1}\sin ax\mathrm{d}x=\sum _{k=0}^{2k+1\le n}(-1{)}^k\frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!}\cos ax+\sum _{k=0}^{2k\le n}(-1{)}^k\frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!}\sin ax$

$\int \frac{\cos ax}{x}\mathrm{d}x=\ln \left|ax\right|+\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{(ax{)}^{2k}}{2k\cdot (2k)!}+C$

$\int \frac{\cos ax}{x^n}\mathrm{d}x=-\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int \frac{\sin ax}{x^{n-1}}\mathrm{d}x\text{(for }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{\cos ax}=\frac{1}{a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{\mathrm{d}x}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{\mathrm{d}x}{{\cos }^{n-2}ax}\text{(for }n>1\text{)}$

$\int \frac{\mathrm{d}x}{1+\cos ax}=\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{\mathrm{d}x}{1-\cos ax}=-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \frac{x\mathrm{d}x}{1+\cos ax}=\frac{x}{a}\tan \frac{ax}{2}+\frac{2}{a^2}\ln |\cos \frac{ax}{2}|+C$

$\int \frac{x\mathrm{d}x}{1-\cos ax}=-\frac{x}{a}\cot \frac{ax}{2}+\frac{2}{a^2}\ln |\sin \frac{ax}{2}|+C$

$\int \frac{\cos ax\mathrm{d}x}{1+\cos ax}=x-\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{\cos ax\mathrm{d}x}{1-\cos ax}=-x-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \cos a_{1}x\cos a_{2}x\mathrm{d}x=\frac{\sin (a_2-a_1)x}{2(a_2-a_1)}+\frac{\sin (a_2+a_1)x}{2(a_2+a_1)}+C\text{(for }\left|a_1\right|\ne \left|a_2\right|\text{)}$

تكاملات مثلثية تحتوي فقط على الظل

$\int \tan ax\mathrm{d}x=-\frac{1}{a}\ln |\cos ax|+C=\frac{1}{a}\ln |\sec ax|+C$

$\int {\tan }^{2}x\mathrm{d}x=\tan x-x+C$

$\int {\tan }^{n}ax\mathrm{d}x=\frac{1}{a(n-1)}{\tan }^{n-1}ax-\int {\tan }^{n-2}ax\mathrm{d}x\text{(for }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{q\tan ax+p}=\frac{1}{p^2+q^2}(px+\frac{q}{a}\ln |q\sin ax+p\cos ax|)+C\text{(for }{p}^2+q^2\ne 0\text{)}$

$\int \frac{\mathrm{d}x}{\tan ax+1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\mathrm{d}x}{\tan ax-1}=-\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\tan ax\mathrm{d}x}{\tan ax+1}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\tan ax\mathrm{d}x}{\tan ax-1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

تكاملات مثلثية تحتوي فقط على ظل التمام

$\int \cot axdx=\frac{1}{a}\ln |\sin ax|+C$

$\int {\cot }^{n}axdx=-\frac{1}{a(n-1)}{\cot }^{n-1}ax-\int {\cot }^{n-2}axdx\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{1+\cot ax}=\int \frac{\tan axdx}{\tan ax+1}$

$\int \frac{dx}{1-\cot ax}=\int \frac{\tan axdx}{\tan ax-1}$

تكاملات مثلثية تحتوي فقط على القاطع

$\int \sec ax\mathrm{d}x=\frac{1}{a}\ln |\sec ax+\tan ax|+C$

$\int {\sec }^{2}x\mathrm{d}x=\tan x+C$

$\int {\sec }^{3}xdx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln |\sec x+\tan x|+C.$

$\int {\sec }^{n}ax\mathrm{d}x=\frac{{\sec }^{n-2}ax\tan ax}{a(n-1)}+\frac{n-2}{n-1}\int {\sec }^{n-2}ax\mathrm{d}x(n\ne 1)$

$\int \frac{\mathrm{d}x}{\sec x+1}=x-\tan \frac{x}{2}+C$

تكاملات مثلثية تحتوي فقط على قاطع التمام

$\int \csc axdx=-\frac{1}{a}\ln |\csc ax+\cot ax|+C=\frac{1}{a}\ln |\csc ax-\cot ax|+C=\frac{1}{a}\ln |\tan \left(\frac{ax}{2}\right)|+C$

$\int {\csc }^{2}x\mathrm{d}x=-\cot x+C$

$\int {\csc }^{n}ax\mathrm{d}x=-\frac{{\csc }^{n-1}\left(ax\right)\cot \left(ax\right)}{a(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}ax\mathrm{d}x(n\ne 1)$

$\int \frac{\mathrm{d}x}{\csc x+1}=x-\frac{2\sin \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}}+C$

$\int \frac{\mathrm{d}x}{\csc x-1}=\frac{2\sin \frac{x}{2}}{\cos \frac{x}{2}-\sin \frac{x}{2}}-x+C$

تكاملات تحتوي على كل من الجيب وجيب التمام

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

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

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

$\int \frac{\cos axdx}{\cos ax+\sin ax}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\cos axdx}{\cos ax-\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax+\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax-\sin ax}=-\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\cos axdx}{(\sin ax)(1+\cos ax)}=-\frac{1}{4a}{\tan }^2\frac{ax}{2}+\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

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

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

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

$\int (\sin ax)(\cos ax)dx=\frac{1}{2a}{\sin }^{}ax+C$

$\int (\sin a_{1}x)(\cos a_{2}x)dx=-\frac{\cos ((a_1-a_2)x)}{2(a_1-a_2)}-\frac{\cos ((a_1+a_2)x)}{2(a_1+a_2)}+C\text{(for }\left|a_1\right|\ne \left|a_2\right|\text{)}$

$\int ({\sin }^{n}ax)(\cos ax)dx=\frac{1}{a(n+1)}{\sin }^{}ax+C\text{(for }n\ne -1\text{)}$

$\int (\sin ax)({\cos }^{}ax)dx=-\frac{1}{a(n+1)}{\cos }^{}ax+C\text{(for }n\ne -1\text{)}$

$\begin{array}{ll}\int ({\sin }^{n}ax)({\cos }^{m}ax)dx& =-\frac{({\sin }^{n-1}ax)({\cos }^{m+1}ax)}{a(n+m)}+\frac{n-1}{n+m}\int ({\sin }^{n-2}ax)({\cos }^{m}ax)dx\text{(for }m,n>0\text{)}\\ & =\frac{({\sin }^{n+1}ax)({\cos }^{m-1}ax)}{a(n+m)}+\frac{m-1}{n+m}\int ({\sin }^{n}ax)({\cos }^{m-2}ax)dx\text{(for }m,n>0\text{)}\end{array}$

$\int \frac{dx}{(\sin ax)(\cos ax)}=\frac{1}{a}\ln |\tan ax|+C$

$\int \frac{dx}{(\sin ax)({\cos }^{}ax)}=\frac{1}{a(n-1){\cos }^{n-1}ax}+\int \frac{dx}{(\sin ax)({\cos }^{}ax)}\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{({\sin }^{n}ax)(\cos ax)}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+\int \frac{dx}{({\sin }^{n-2}ax)(\cos ax)}\text{(for }n\ne 1\text{)}$

$\int \frac{\sin axdx}{{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

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

$\int \frac{{\sin }^{2}axdx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{\cos ax}=-\frac{{\sin }^{n-1}ax}{a(n-1)}+\int \frac{{\sin }^{n-2}axdx}{\cos ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\{\begin{array}{cc}\frac{{\sin }^{n+1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}axdx}{{\cos }^{m-2}ax}& \text{(for }m\ne 1\text{)}\\ \frac{{\sin }^{n-1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m-2}ax}& \text{(for }m\ne 1\text{)}\\ -\frac{{\sin }^{n-1}ax}{a(n-m){\cos }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m}ax}& \text{(for }m\ne n\text{)}\end{array}$

$\int \frac{\cos axdx}{{\sin }^{n}ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{2}axdx}{\sin ax}=\frac{1}{a}(\cos ax+\ln |\tan \frac{ax}{2}|)+C$

$\int \frac{{\cos }^{2}axdx}{{\sin }^{n}ax}=-\frac{1}{n-1}(\frac{\cos ax}{a{\sin }^{n-1}ax}+\int \frac{dx}{{\sin }^{n-2}ax})\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=\{\begin{array}{cc}-\frac{{\cos }^{n+1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\cos }^{n}axdx}{{\sin }^{m-2}ax}& \text{(for }m\ne 1\text{)}\\ -\frac{{\cos }^{n-1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m-2}ax}& \text{(for }m\ne 1\text{)}\\ \frac{{\cos }^{n-1}ax}{a(n-m){\sin }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m}ax}& \text{(for }m\ne n\text{)}\end{array}$

تكاملات تحتوي على كل من الجيب والظل

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

$\int \frac{{\tan }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n-1)}{\tan }^{n-1}(ax)+C\text{(for }n\ne 1\text{)}$

تكاملات تحتوي على كل من جيب التمام والظل

$\int \frac{{\tan }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(n+1)}{\tan }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

تكاملات تحتوي على كل من الجيب وظل التمام

$\int \frac{{\cot }^{n}axdx}{{\sin }^{2}ax}=-\frac{1}{a(n+1)}{\cot }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

تكاملات تحتوي على كل من جيب التمام وظل التمام

$\int \frac{{\cot }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(1-n)}{\tan }^{1-n}ax+C\text{(for }n\ne 1\text{)}$

تكاملات تحتوي على كل من القاطع والظل

$\int (\sec x)(\tan x)dx=\sec x+C$

تكاملات تحتوي على كل من ظل التمام وقاطع التمام

$\int (\csc x)(\cot x)dx=-\csc x+C$