1. تحويل نطقب:حساب المثلثات

تكاملات مثلثية تحتوي فقط على الجيب (جا)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Antiderivatives containing only cosine

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Antiderivatives containing only tangent

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

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

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

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

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

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

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

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

Antiderivatives containing only secant

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

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

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

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

Antiderivatives containing only cosecant

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

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

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

Antiderivatives containing only cotangent

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

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

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

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

Antiderivatives containing both sine and cosine

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Antiderivatives containing both sine and tangent

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

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

Antiderivatives containing both cosine and tangent

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

Antiderivatives containing both sine and cotangent

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

Antiderivatives containing both cosine and cotangent

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

Antiderivatives with symmetric limits

${\displaystyle \int _{-c ^{c \sin {x \;dx=0\!$

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

${\displaystyle \int _{-c ^{c \tan {x \;dx=0\!$

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