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 ال[[جيب (رياضيات)|جيب]]


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

: \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\!

: \int\sin a_1x\sin a_2x\;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{)}\,\!

: \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{)}\,\!

: \int\frac{dx}{\sin ax} = \frac{1}{a}\ln \left|\tan\frac{ax}{2}\right|+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} \qquad\mbox{(for }n>1\mbox{)}\,\!

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

: \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{)}\,\!

: \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{)}\,\!

: \int\frac{\sin ax}{x} dx = \sum_{n=0}^\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\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)+C

: \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

: \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

: \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


## 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 \qquad\mbox{(for }n>0\mbox{)}\,\!

: \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} + \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \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\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq 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|ax|+\sum_{k=1}^\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 \qquad\mbox{(for }n\neq 1\mbox{)}\,\!

: \int\frac{\mathrm{d}x}{\cos ax} = \frac{1}{a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+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} \qquad\mbox{(for }n>1\mbox{)}\,\!

: \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\left|\cos\frac{ax}{2}\right|+C

: \int\frac{x\;\mathrm{d}x}{1-\cos ax} = -\frac{x}{a}\cot\frac{ax}{2}+\frac{2}{a^2}\ln\left|\sin\frac{ax}{2}\right|+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_1x\cos a_2x\;\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 \qquad\mbox{(for }|a_1|\neq|a_2|\mbox{)}\,\!



## تكاملات مثلثية تحتوي فقط على ال[[ظل (حساب المثلثات)|ظل]]


: \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 \qquad\mbox{(for }n\neq 1\mbox{)}\,\!

: \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 \qquad\mbox{(for }p^2 + q^2\neq 0\mbox{)}\,\!

: \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 ax\,dx = \frac{1}{a}\ln|\sin ax|+C

: \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{)}

: \int\frac{dx}{1 + \cot ax} = \int\frac{\tan ax\,dx}{\tan ax+1}

: \int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\,dx}{\tan ax-1}



## تكاملات مثلثية تحتوي فقط على ال[[قاطع (حساب المثلثات)|قاطع]]


:\int \sec{ax} \, \mathrm{d}x = \frac{1}{a}\ln{\left| \sec{ax} + \tan{ax}\right|}+C

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

:\int \sec^3 x \, dx = \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 \qquad (n \neq 1)\,\!

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



## تكاملات مثلثية تحتوي فقط على [[قاطع التمام]]


:\int \csc{ax} \, dx= -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C= \frac{1}{a}\ln{\left| \csc{ax}-\cot{ax}\right|}+C = \frac{1}{a}\ln{\left| \tan{\left( \frac{ax}{2} \right)}\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 \qquad (n\neq1)\,\!

:\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\left|\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right)\right|+C

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

: \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)

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

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

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

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

: \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

: \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

: \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

: \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

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

: \int(\sin a_1x)(\cos a_2x)\,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{)}

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

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

: \begin{align}
\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{)} \\
&= \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{)}
\end{align}

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

: \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{)}

: \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{)}

: \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{)}

: \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

: \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{)}

: \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{)}

: \int\frac{\sin^n ax\,dx}{\cos^m ax} = \begin{cases}
\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} &\mbox{(for }m\neq 1\mbox{)} \\
\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} &\mbox{(for }m\neq 1\mbox{)} \\
-\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} &\mbox{(for }m\neq n\mbox{)}
\end{cases}

: \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{)}

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

: \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{)}

: \int\frac{\cos^n ax\,dx}{\sin^m ax} = \begin{cases}
-\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} &\mbox{(for }m\neq 1\mbox{)} \\
-\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} &\mbox{(for }m\neq 1\mbox{)} \\
\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} &\mbox{(for }m\neq n\mbox{)}
\end{cases}


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


## تكاملات تحتوي على كل من جيب التمام والظل


: \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{)}


## تكاملات تحتوي على كل من الجيب وظل التمام


: \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{)}


## تكاملات تحتوي على كل من جيب التمام وظل التمام


: \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{)}


## 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


## أنظر أيضا
* تكامل مثلثي

تصنيف:تكاملات
تصنيف:دوال مثلثية
تصنيف:حساب المثلثات
تصنيف:قوائم متعلقة بالرياضيات
مثلثية