घर
बेतरतीब
निकट
लॉग-इन करें
सेटिंग्स
दान करें
विकिपीडिया के बारे में
अस्वीकरण
खोजें
त्रिकोणमितीय फलनों के समाकलों की सूची
भाषा
ध्यान रखें
सम्पादित करें
यहाँ
त्रिकोणमितीय फलनों
के समाकलों (इंटीग्रल्स) की सूची दी गयी है।
अनुक्रम
1
वे फलन जिनमें केवल साइन (sine) है
2
Integrands involving only cosine
3
Integrands involving only tangent
4
Integrands involving only secant
5
Integrands involving only cosecant
6
Integrands involving only cotangent
7
Integrands involving both sine and cosine
8
Integrands involving both sine and tangent
9
Integrand involving both cosine and tangent
10
Integrand involving both sine and cotangent
11
Integrand involving both cosine and cotangent
12
Integrand involving both secant and tangent
13
Integrals in a quarter period
14
Integrals with symmetric limits
15
Integral over a full circle
16
सन्दर्भ
17
इन्हें भी देखें
वे फलन जिनमें केवल साइन (sine) है
संपादित करें
∫
sin
a
x
d
x
=
−
1
a
cos
a
x
+
C
{\displaystyle \int \sin ax\;\mathrm {d} x=-{\frac {1}{a}}\cos ax+C\,\!}
∫
sin
2
a
x
d
x
=
x
2
−
1
4
a
sin
2
a
x
+
C
=
x
2
−
1
2
a
sin
a
x
cos
a
x
+
C
{\displaystyle \int \sin ^{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\!}
∫
sin
3
a
x
d
x
=
cos
3
a
x
12
a
−
3
cos
a
x
4
a
+
C
{\displaystyle \int \sin ^{3}{ax}\;\mathrm {d} x={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C\!}
∫
x
sin
2
a
x
d
x
=
x
2
4
−
x
4
a
sin
2
a
x
−
1
8
a
2
cos
2
a
x
+
C
{\displaystyle \int x\sin ^{2}{ax}\;\mathrm {d} x={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C\!}
∫
x
2
sin
2
a
x
d
x
=
x
3
6
−
(
x
2
4
a
−
1
8
a
3
)
sin
2
a
x
−
x
4
a
2
cos
2
a
x
+
C
{\displaystyle \int x^{2}\sin ^{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\!}
∫
sin
b
1
x
sin
b
2
x
d
x
=
sin
(
(
b
2
−
b
1
)
x
)
2
(
b
2
−
b
1
)
−
sin
(
(
b
1
+
b
2
)
x
)
2
(
b
1
+
b
2
)
+
C
(for
|
b
1
|
≠
|
b
2
|
)
{\displaystyle \int \sin b_{1}x\sin b_{2}x\;\mathrm {d} x={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(for }}|b_{1}|\neq |b_{2}|{\mbox{)}}\,\!}
∫
sin
n
a
x
d
x
=
−
sin
n
−
1
a
x
cos
a
x
n
a
+
n
−
1
n
∫
sin
n
−
2
a
x
d
x
(for
n
>
0
)
{\displaystyle \int \sin ^{n}{ax}\;\mathrm {d} x=-{\frac {\sin ^{n-1}ax\cos ax}{na}}+{\frac {n-1}{n}}\int \sin ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}
∫
d
x
sin
a
x
=
−
1
a
ln
|
csc
a
x
+
cot
a
x
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ax}}=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}
∫
d
x
sin
n
a
x
=
cos
a
x
a
(
1
−
n
)
sin
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
sin
n
−
2
a
x
(for
n
>
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{n}ax}}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}
∫
x
sin
a
x
d
x
=
sin
a
x
a
2
−
x
cos
a
x
a
+
C
{\displaystyle \int x\sin ax\;\mathrm {d} x={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C\,\!}
∫
x
n
sin
a
x
d
x
=
−
x
n
a
cos
a
x
+
n
a
∫
x
n
−
1
cos
a
x
d
x
=
∑
k
=
0
2
k
≤
n
(
−
1
)
k
+
1
x
n
−
2
k
a
1
+
2
k
n
!
(
n
−
2
k
)
!
cos
a
x
+
∑
k
=
0
2
k
+
1
≤
n
(
−
1
)
k
x
n
−
1
−
2
k
a
2
+
2
k
n
!
(
n
−
2
k
−
1
)
!
sin
a
x
(for
n
>
0
)
{\displaystyle \int x^{n}\sin ax\;\mathrm {d} x=-{\frac {x^{n}}{a}}\cos ax+{\frac {n}{a}}\int x^{n-1}\cos ax\;\mathrm {d} x=\sum _{k=0}^{2k\leq n}(-1)^{k+1}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\cos ax+\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-1-2k}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\sin ax\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}
∫
sin
a
x
x
d
x
=
∑
n
=
0
∞
(
−
1
)
n
(
a
x
)
2
n
+
1
(
2
n
+
1
)
⋅
(
2
n
+
1
)
!
+
C
{\displaystyle \int {\frac {\sin ax}{x}}\mathrm {d} x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C\,\!}
∫
sin
a
x
x
n
d
x
=
−
sin
a
x
(
n
−
1
)
x
n
−
1
+
a
n
−
1
∫
cos
a
x
x
n
−
1
d
x
{\displaystyle \int {\frac {\sin ax}{x^{n}}}\mathrm {d} x=-{\frac {\sin ax}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}\int {\frac {\cos ax}{x^{n-1}}}\mathrm {d} x\,\!}
∫
d
x
1
±
sin
a
x
=
1
a
tan
(
a
x
2
∓
π
4
)
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}
∫
x
d
x
1
+
sin
a
x
=
x
a
tan
(
a
x
2
−
π
4
)
+
2
a
2
ln
|
cos
(
a
x
2
−
π
4
)
|
+
C
{\displaystyle \int {\frac {x\;\mathrm {d} x}{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}
∫
x
d
x
1
−
sin
a
x
=
x
a
cot
(
π
4
−
a
x
2
)
+
2
a
2
ln
|
sin
(
π
4
−
a
x
2
)
|
+
C
{\displaystyle \int {\frac {x\;\mathrm {d} x}{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}
∫
sin
a
x
d
x
1
±
sin
a
x
=
±
x
+
1
a
tan
(
π
4
∓
a
x
2
)
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}
Integrands involving only
cosine
संपादित करें
∫
cos
a
x
d
x
=
1
a
sin
a
x
+
C
{\displaystyle \int \cos ax\;\mathrm {d} x={\frac {1}{a}}\sin ax+C\,\!}
∫
cos
2
a
x
d
x
=
x
2
+
1
4
a
sin
2
a
x
+
C
=
x
2
+
1
2
a
sin
a
x
cos
a
x
+
C
{\displaystyle \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\!}
∫
cos
n
a
x
d
x
=
cos
n
−
1
a
x
sin
a
x
n
a
+
n
−
1
n
∫
cos
n
−
2
a
x
d
x
(for
n
>
0
)
{\displaystyle \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{)}}\,\!}
∫
x
cos
a
x
d
x
=
cos
a
x
a
2
+
x
sin
a
x
a
+
C
{\displaystyle \int x\cos ax\;\mathrm {d} x={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C\,\!}
∫
x
2
cos
2
a
x
d
x
=
x
3
6
+
(
x
2
4
a
−
1
8
a
3
)
sin
2
a
x
+
x
4
a
2
cos
2
a
x
+
C
{\displaystyle \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\!}
∫
x
n
cos
a
x
d
x
=
x
n
sin
a
x
a
−
n
a
∫
x
n
−
1
sin
a
x
d
x
=
∑
k
=
0
2
k
+
1
≤
n
(
−
1
)
k
x
n
−
2
k
−
1
a
2
+
2
k
n
!
(
n
−
2
k
−
1
)
!
cos
a
x
+
∑
k
=
0
2
k
≤
n
(
−
1
)
k
x
n
−
2
k
a
1
+
2
k
n
!
(
n
−
2
k
)
!
sin
a
x
{\displaystyle \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\!}
∫
cos
a
x
x
d
x
=
ln
|
a
x
|
+
∑
k
=
1
∞
(
−
1
)
k
(
a
x
)
2
k
2
k
⋅
(
2
k
)
!
+
C
{\displaystyle \int {\frac {\cos ax}{x}}\mathrm {d} x=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C\,\!}
∫
cos
a
x
x
n
d
x
=
−
cos
a
x
(
n
−
1
)
x
n
−
1
−
a
n
−
1
∫
sin
a
x
x
n
−
1
d
x
(for
n
≠
1
)
{\displaystyle \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{)}}\,\!}
∫
d
x
cos
a
x
=
1
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
∫
d
x
cos
n
a
x
=
sin
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
cos
n
−
2
a
x
(for
n
>
1
)
{\displaystyle \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{)}}\,\!}
∫
d
x
1
+
cos
a
x
=
1
a
tan
a
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}
∫
d
x
1
−
cos
a
x
=
−
1
a
cot
a
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}
∫
x
d
x
1
+
cos
a
x
=
x
a
tan
a
x
2
+
2
a
2
ln
|
cos
a
x
2
|
+
C
{\displaystyle \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}
∫
x
d
x
1
−
cos
a
x
=
−
x
a
cot
a
x
2
+
2
a
2
ln
|
sin
a
x
2
|
+
C
{\displaystyle \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}
∫
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
2
−
a
1
)
x
)
2
(
a
2
−
a
1
)
+
sin
(
(
a
2
+
a
1
)
x
)
2
(
a
2
+
a
1
)
+
C
(for
|
a
1
|
≠
|
a
2
|
)
{\displaystyle \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\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}
Integrands involving only
tangent
संपादित करें
∫
tan
a
x
d
x
=
−
1
a
ln
|
cos
a
x
|
+
C
=
1
a
ln
|
sec
a
x
|
+
C
{\displaystyle \int \tan ax\;\mathrm {d} x=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C\,\!}
∫
tan
2
x
d
x
=
tan
x
−
x
+
C
{\displaystyle \int \tan ^{2}{x}\,\mathrm {d} x=\tan {x}-x+C}
∫
tan
n
a
x
d
x
=
1
a
(
n
−
1
)
tan
n
−
1
a
x
−
∫
tan
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \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{)}}\,\!}
∫
d
x
q
tan
a
x
+
p
=
1
p
2
+
q
2
(
p
x
+
q
a
ln
|
q
sin
a
x
+
p
cos
a
x
|
)
+
C
(for
p
2
+
q
2
≠
0
)
{\displaystyle \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{)}}\,\!}
∫
d
x
tan
a
x
+
1
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\tan ax+1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}
∫
d
x
tan
a
x
−
1
=
−
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\tan ax-1}}=-{\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}
∫
tan
a
x
d
x
tan
a
x
+
1
=
x
2
−
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\tan ax\;\mathrm {d} x}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}
∫
tan
a
x
d
x
tan
a
x
−
1
=
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\tan ax\;\mathrm {d} x}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}
Integrands involving only
secant
संपादित करें
See
Integral of the secant function
.
∫
sec
a
x
d
x
=
1
a
ln
|
sec
a
x
+
tan
a
x
|
+
C
{\displaystyle \int \sec {ax}\,\mathrm {d} x={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}
∫
sec
2
x
d
x
=
tan
x
+
C
{\displaystyle \int \sec ^{2}{x}\,\mathrm {d} x=\tan {x}+C}
∫
sec
3
x
d
x
=
1
2
sec
x
tan
x
+
1
2
ln
|
sec
x
+
tan
x
|
+
C
.
{\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C.}
∫
sec
n
a
x
d
x
=
sec
n
−
2
a
x
tan
a
x
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
sec
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \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 {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
sec
x
+
1
=
x
−
tan
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}
Integrands involving only
cosecant
संपादित करें
∫
csc
(
a
x
)
d
x
=
−
1
a
ln
|
csc
a
x
+
cot
a
x
|
+
C
{\displaystyle \int \csc(ax)\mathrm {d} x=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}
∫
csc
2
x
d
x
=
−
cot
x
+
C
{\displaystyle \int \csc ^{2}{x}\,\mathrm {d} x=-\cot {x}+C}
∫
csc
n
a
x
d
x
=
−
csc
n
−
1
(
a
x
)
cos
(
a
x
)
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
csc
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \csc ^{n}{ax}\,\mathrm {d} x=-{\frac {\csc ^{n-1}\left(ax\right)\cos \left(ax\right)}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,\mathrm {d} x\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
csc
x
+
1
=
x
−
2
sin
x
2
cos
x
2
+
sin
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\csc {x}+1}}=x-{\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}}+C}
∫
d
x
csc
x
−
1
=
2
sin
x
2
cos
x
2
−
sin
x
2
−
x
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\csc {x}-1}}={\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}-x+C}
Integrands involving only
cotangent
संपादित करें
∫
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
(for
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{(for }}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}}\,\!}
Integrands involving both
sine
and
cosine
संपादित करें
∫
d
x
cos
a
x
±
sin
a
x
=
1
a
2
ln
|
tan
(
a
x
2
±
π
8
)
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\cos ax\pm \sin ax}}={\frac {1}{a{\sqrt {2}}}}\ln \left|\tan \left({\frac {ax}{2}}\pm {\frac {\pi }{8}}\right)\right|+C}
∫
d
x
(
cos
a
x
±
sin
a
x
)
2
=
1
2
a
tan
(
a
x
∓
π
4
)
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C}
∫
d
x
(
cos
x
+
sin
x
)
n
=
1
n
−
1
(
sin
x
−
cos
x
(
cos
x
+
sin
x
)
n
−
1
−
2
(
n
−
2
)
∫
d
x
(
cos
x
+
sin
x
)
n
−
2
)
{\displaystyle \int {\frac {\mathrm {d} x}{(\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 {\mathrm {d} x}{(\cos x+\sin x)^{n-2}}}\right)}
∫
cos
a
x
d
x
cos
a
x
+
sin
a
x
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
∫
cos
a
x
d
x
cos
a
x
−
sin
a
x
=
x
2
−
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}
∫
sin
a
x
d
x
cos
a
x
+
sin
a
x
=
x
2
−
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
∫
sin
a
x
d
x
cos
a
x
−
sin
a
x
=
−
x
2
−
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}
∫
cos
a
x
d
x
sin
a
x
(
1
+
cos
a
x
)
=
−
1
4
a
tan
2
a
x
2
+
1
2
a
ln
|
tan
a
x
2
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\sin ax(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}
∫
cos
a
x
d
x
sin
a
x
(
1
−
cos
a
x
)
=
−
1
4
a
cot
2
a
x
2
−
1
2
a
ln
|
tan
a
x
2
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\sin ax(1-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}
∫
sin
a
x
d
x
cos
a
x
(
1
+
sin
a
x
)
=
1
4
a
cot
2
(
a
x
2
+
π
4
)
+
1
2
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\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}
∫
sin
a
x
d
x
cos
a
x
(
1
−
sin
a
x
)
=
1
4
a
tan
2
(
a
x
2
+
π
4
)
−
1
2
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\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}
∫
sin
a
x
cos
a
x
d
x
=
−
1
2
a
cos
2
a
x
+
C
{\displaystyle \int \sin ax\cos ax\;\mathrm {d} x=-{\frac {1}{2a}}\cos ^{2}ax+C\,\!}
∫
sin
a
1
x
cos
a
2
x
d
x
=
−
cos
(
(
a
1
−
a
2
)
x
)
2
(
a
1
−
a
2
)
−
cos
(
(
a
1
+
a
2
)
x
)
2
(
a
1
+
a
2
)
+
C
(for
|
a
1
|
≠
|
a
2
|
)
{\displaystyle \int \sin a_{1}x\cos a_{2}x\;\mathrm {d} x=-{\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{)}}\,\!}
∫
sin
n
a
x
cos
a
x
d
x
=
1
a
(
n
+
1
)
sin
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int \sin ^{n}ax\cos ax\;\mathrm {d} x={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
∫
sin
a
x
cos
n
a
x
d
x
=
−
1
a
(
n
+
1
)
cos
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int \sin ax\cos ^{n}ax\;\mathrm {d} x=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
∫
sin
n
a
x
cos
m
a
x
d
x
=
−
sin
n
−
1
a
x
cos
m
+
1
a
x
a
(
n
+
m
)
+
n
−
1
n
+
m
∫
sin
n
−
2
a
x
cos
m
a
x
d
x
(for
m
,
n
>
0
)
{\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;\mathrm {d} x=-{\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\;\mathrm {d} x\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}
also:
∫
sin
n
a
x
cos
m
a
x
d
x
=
sin
n
+
1
a
x
cos
m
−
1
a
x
a
(
n
+
m
)
+
m
−
1
n
+
m
∫
sin
n
a
x
cos
m
−
2
a
x
d
x
(for
m
,
n
>
0
)
{\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;\mathrm {d} x={\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\;\mathrm {d} x\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}
∫
d
x
sin
a
x
cos
a
x
=
1
a
ln
|
tan
a
x
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ax\cos ax}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}
∫
d
x
sin
a
x
cos
n
a
x
=
1
a
(
n
−
1
)
cos
n
−
1
a
x
+
∫
d
x
sin
a
x
cos
n
−
2
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ax\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+\int {\frac {\mathrm {d} x}{\sin ax\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
sin
n
a
x
cos
a
x
=
−
1
a
(
n
−
1
)
sin
n
−
1
a
x
+
∫
d
x
sin
n
−
2
a
x
cos
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{n}ax\cos ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
a
x
d
x
cos
n
a
x
=
1
a
(
n
−
1
)
cos
n
−
1
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
2
a
x
d
x
cos
a
x
=
−
1
a
sin
a
x
+
1
a
ln
|
tan
(
π
4
+
a
x
2
)
|
+
C
{\displaystyle \int {\frac {\sin ^{2}ax\;\mathrm {d} x}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}
∫
sin
2
a
x
d
x
cos
n
a
x
=
sin
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
−
1
n
−
1
∫
d
x
cos
n
−
2
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\sin ^{2}ax\;\mathrm {d} x}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
n
a
x
d
x
cos
a
x
=
−
sin
n
−
1
a
x
a
(
n
−
1
)
+
∫
sin
n
−
2
a
x
d
x
cos
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\;\mathrm {d} x}{\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
n
a
x
d
x
cos
m
a
x
=
sin
n
+
1
a
x
a
(
m
−
1
)
cos
m
−
1
a
x
−
n
−
m
+
2
m
−
1
∫
sin
n
a
x
d
x
cos
m
−
2
a
x
(for
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-m+2}{m-1}}\int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}
also:
∫
sin
n
a
x
d
x
cos
m
a
x
=
−
sin
n
−
1
a
x
a
(
n
−
m
)
cos
m
−
1
a
x
+
n
−
1
n
−
m
∫
sin
n
−
2
a
x
d
x
cos
m
a
x
(for
m
≠
n
)
{\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\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\;\mathrm {d} x}{\cos ^{m}ax}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}
also:
∫
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
(for
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{(for }}m\neq 1{\mbox{)}}\,\!}
∫
cos
a
x
d
x
sin
n
a
x
=
−
1
a
(
n
−
1
)
sin
n
−
1
a
x
+
C
(for
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{(for }}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}
∫
cos
2
a
x
d
x
sin
n
a
x
=
−
1
n
−
1
(
cos
a
x
a
sin
n
−
1
a
x
)
+
∫
d
x
sin
n
−
2
a
x
)
(for
n
≠
1
)
{\displaystyle \int {\frac {\cos ^{2}ax\;\mathrm {d} x}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax)}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
cos
n
a
x
d
x
sin
m
a
x
=
−
cos
n
+
1
a
x
a
(
m
−
1
)
sin
m
−
1
a
x
−
n
−
m
+
2
m
−
1
∫
cos
n
a
x
d
x
sin
m
−
2
a
x
(for
m
≠
1
)
{\displaystyle \int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\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\;\mathrm {d} x}{\sin ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}
also:
∫
cos
n
a
x
d
x
sin
m
a
x
=
cos
n
−
1
a
x
a
(
n
−
m
)
sin
m
−
1
a
x
+
n
−
1
n
−
m
∫
cos
n
−
2
a
x
d
x
sin
m
a
x
(for
m
≠
n
)
{\displaystyle \int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\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\;\mathrm {d} x}{\sin ^{m}ax}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}
also:
∫
cos
n
a
x
d
x
sin
m
a
x
=
−
cos
n
−
1
a
x
a
(
m
−
1
)
sin
m
−
1
a
x
−
n
−
1
m
−
1
∫
cos
n
−
2
a
x
d
x
sin
m
−
2
a
x
(for
m
≠
1
)
{\displaystyle \int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\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\;\mathrm {d} x}{\sin ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}
Integrands involving both
sine
and
tangent
संपादित करें
∫
sin
a
x
tan
a
x
d
x
=
1
a
(
ln
|
sec
a
x
+
tan
a
x
|
−
sin
a
x
)
+
C
{\displaystyle \int \sin ax\tan ax\;\mathrm {d} x={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}
∫
tan
n
a
x
d
x
sin
2
a
x
=
1
a
(
n
−
1
)
tan
n
−
1
(
a
x
)
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\tan ^{n}ax\;\mathrm {d} x}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
Integrand involving both
cosine
and
tangent
संपादित करें
∫
tan
n
a
x
d
x
cos
2
a
x
=
1
a
(
n
+
1
)
tan
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int {\frac {\tan ^{n}ax\;\mathrm {d} x}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
Integrand involving both
sine
and
cotangent
संपादित करें
∫
cot
n
a
x
d
x
sin
2
a
x
=
−
1
a
(
n
+
1
)
cot
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int {\frac {\cot ^{n}ax\;\mathrm {d} x}{\sin ^{2}ax}}=-{\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
Integrand involving both
cosine
and
cotangent
संपादित करें
∫
cot
n
a
x
d
x
cos
2
a
x
=
1
a
(
1
−
n
)
tan
1
−
n
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\cot ^{n}ax\;\mathrm {d} x}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
Integrand involving both
secant
and
tangent
संपादित करें
∫
sec
x
tan
x
d
x
=
sec
x
+
C
{\displaystyle \int \sec x\tan x\;\mathrm {d} x=\sec x+C}
Integrals in a quarter period
संपादित करें
∫
0
π
2
sin
n
x
d
x
=
∫
0
π
2
cos
n
x
d
x
=
{
n
−
1
n
⋅
n
−
3
n
−
2
⋯
3
4
⋅
1
2
⋅
π
2
,
if
n
is even
n
−
1
n
⋅
n
−
3
n
−
2
⋯
4
5
⋅
2
3
if
n
is odd and more than 1
{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\begin{cases}{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {3}{4}}\cdot {\frac {1}{2}}\cdot {\frac {\pi }{2}},&{\text{if }}n{\text{ is even}}\\{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {4}{5}}\cdot {\frac {2}{3}}&{\text{if }}n{\text{ is odd and more than 1}}\end{cases}}}
Integrals with symmetric limits
संपादित करें
∫
−
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
(for
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{(for }}n=1,3,5...{\mbox{)}}\,\!}
∫
−
a
2
a
2
x
2
sin
2
n
π
x
a
d
x
=
a
3
(
n
2
π
2
−
6
(
−
1
)
n
)
24
n
2
π
2
=
a
3
24
(
1
−
6
(
−
1
)
n
n
2
π
2
)
(for
n
=
1
,
2
,
3
,
.
.
.
)
{\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\;\mathrm {d} x={\frac {a^{3}(n^{2}\pi ^{2}-6(-1)^{n})}{24n^{2}\pi ^{2}}}={\frac {a^{3}}{24}}(1-6{\frac {(-1)^{n}}{n^{2}\pi ^{2}}})\qquad {\mbox{(for }}n=1,2,3,...{\mbox{)}}\,\!}
Integral over a full circle
संपादित करें
∫
0
2
π
sin
2
m
+
1
x
cos
2
n
+
1
x
d
x
=
0
{
n
,
m
}
∈
Z
{\displaystyle \int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{2n+1}{x}\;\mathrm {d} x=0\!\qquad \{n,m\}\in \mathbb {Z} }
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