C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.
These formulas only state in another form the assertions in the table of derivatives .
sửa
more integrals: List of integrals of rational functions
∫
a
d
x
=
a
x
+
C
{\displaystyle \int a\,dx=ax+C}
∫
x
a
d
x
=
x
a
+
1
a
+
1
+
C
{\displaystyle \int x^{a}\,dx={\frac {x^{a+1}}{a+1}}+C}
∫
1
x
d
x
=
ln
|
x
|
+
C
{\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}
sửa
more integrals: List of integrals of exponential functions
∫
e
x
d
x
=
e
x
+
C
{\displaystyle \int e^{x}\,dx=e^{x}+C}
∫
a
x
d
x
=
a
x
ln
a
+
C
{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C}
sửa
more integrals: List of integrals of logarithmic functions
∫
ln
x
d
x
=
x
|
l
n
x
−
x
|
+
C
{\displaystyle \int \ln x\,dx=x|lnx-x|+C}
∫
log
a
x
d
x
=
x
log
a
x
−
x
ln
a
+
C
{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C}
sửa
more integrals: List of integrals of trigonometric functions
∫
sin
x
d
x
=
−
cos
x
+
C
{\displaystyle \int \sin {x}\,dx=-\cos {x}+C}
∫
cos
x
d
x
=
sin
x
+
C
{\displaystyle \int \cos {x}\,dx=\sin {x}+C}
∫
tan
x
d
x
=
−
ln
|
cos
x
|
+
C
=
ln
|
sec
x
|
+
C
{\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C=\ln {\left|\sec {x}\right|}+C}
∫
cot
x
d
x
=
ln
|
sin
x
|
+
C
{\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}
∫
sec
x
d
x
=
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}
∫
csc
x
d
x
=
−
ln
|
csc
x
+
cot
x
|
+
C
{\displaystyle \int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C}
∫
sec
2
x
d
x
=
tan
x
+
C
{\displaystyle \int \sec ^{2}x\,dx=\tan x+C}
∫
csc
2
x
d
x
=
−
cot
x
+
C
{\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}
∫
sec
x
tan
x
d
x
=
sec
x
+
C
{\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}
∫
csc
x
cot
x
d
x
=
−
csc
x
+
C
{\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}
∫
sin
2
x
d
x
=
1
2
(
x
−
sin
2
x
2
)
+
C
=
1
2
(
x
−
sin
x
cos
x
)
+
C
{\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C}
∫
cos
2
x
d
x
=
1
2
(
x
+
sin
2
x
2
)
+
C
=
1
2
(
x
+
sin
x
cos
x
)
+
C
{\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos 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}
(see integral of secant cubed )
∫
sin
n
x
d
x
=
−
sin
n
−
1
x
cos
x
n
+
n
−
1
n
∫
sin
n
−
2
x
d
x
{\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}
∫
cos
n
x
d
x
=
cos
n
−
1
x
sin
x
n
+
n
−
1
n
∫
cos
n
−
2
x
d
x
{\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}
sửa
more integrals: List of integrals of inverse trigonometric functions
∫
arcsin
x
d
x
=
x
arcsin
x
+
1
−
x
2
+
C
{\displaystyle \int \arcsin {x}\,dx=x\,\arcsin {x}+{\sqrt {1-x^{2}}}+C}
∫
arccos
x
d
x
=
x
arccos
x
−
1
−
x
2
+
C
{\displaystyle \int \arccos {x}\,dx=x\,\arccos {x}-{\sqrt {1-x^{2}}}+C}
∫
arctan
x
d
x
=
x
arctan
x
−
1
2
ln
|
1
+
x
2
|
+
C
{\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}
∫
arcsec
x
d
x
=
x
arcsec
x
+
x
2
−
1
ln
(
x
+
x
2
−
1
)
x
1
−
1
x
2
+
C
{\displaystyle \int \operatorname {arcsec} {x}\,dx=x\,\operatorname {arcsec} {x}+{\frac {{\sqrt {x^{2}-1}}\ln {(x+{\sqrt {x^{2}-1}})}}{x\,{\sqrt {1-{\frac {1}{x^{2}}}}}}}+C}
∫
arccsc
x
d
x
=
x
arccsc
x
+
x
2
−
1
ln
(
x
+
x
2
−
1
)
x
1
−
1
x
2
+
C
{\displaystyle \int \operatorname {arccsc} {x}\,dx=x\,\operatorname {arccsc} {x}+{\frac {{\sqrt {x^{2}-1}}\ln {(x+{\sqrt {x^{2}-1}})}}{x\,{\sqrt {1-{\frac {1}{x^{2}}}}}}}+C}
∫
arccot
x
d
x
=
x
arccot
x
+
1
2
ln
|
1
+
x
2
|
+
C
{\displaystyle \int \operatorname {arccot} {x}\,dx=x\,\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}
sửa
more integrals: List of integrals of hyperbolic functions
∫
sinh
x
d
x
=
cosh
x
+
C
{\displaystyle \int \sinh x\,dx=\cosh x+C}
∫
cosh
x
d
x
=
sinh
x
+
C
{\displaystyle \int \cosh x\,dx=\sinh x+C}
∫
tanh
x
d
x
=
ln
|
cosh
x
|
+
C
{\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}
∫
cosech
x
d
x
=
ln
|
tanh
x
2
|
+
C
{\displaystyle \int {\mbox{cosech}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}
∫
sech
x
d
x
=
arctan
(
sinh
x
)
+
C
{\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan \,(\sinh x)+C}
∫
coth
x
d
x
=
ln
|
sinh
x
|
+
C
{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}
∫
sech
2
x
d
x
=
tanh
x
+
C
{\displaystyle \int {\mbox{sech}}^{2}x\,dx=\tanh x+C}
sửa
more integrals: List of integrals of inverse hyperbolic functions
∫
arsinh
x
d
x
=
x
arsinh
x
−
x
2
+
1
+
C
{\displaystyle \int \operatorname {arsinh} \,x\,dx=x\,\operatorname {arsinh} \,x-{\sqrt {x^{2}+1}}+C}
∫
arcosh
x
d
x
=
x
arcosh
x
−
x
2
−
1
+
C
{\displaystyle \int \operatorname {arcosh} \,x\,dx=x\,\operatorname {arcosh} \,x-{\sqrt {x^{2}-1}}+C}
∫
artanh
x
d
x
=
x
artanh
x
+
1
2
ln
(
1
−
x
2
)
+
C
{\displaystyle \int \operatorname {artanh} \,x\,dx=x\,\operatorname {artanh} \,x+{\frac {1}{2}}\ln {(1-x^{2})}+C}
∫
arcsch
x
d
x
=
x
arcsch
x
+
ln
[
x
(
1
+
1
x
2
+
1
)
]
+
C
{\displaystyle \int \operatorname {arcsch} \,x\,dx=x\,\operatorname {arcsch} \,x+\ln {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C}
∫
arsech
x
d
x
=
x
arsech
x
−
arctan
(
x
x
−
1
1
−
x
1
+
x
)
+
C
{\displaystyle \int \operatorname {arsech} \,x\,dx=x\,\operatorname {arsech} \,x-\arctan {\left({\frac {x}{x-1}}{\sqrt {\frac {1-x}{1+x}}}\right)}+C}
∫
arcoth
x
d
x
=
x
arcoth
x
+
1
2
ln
(
x
2
−
1
)
+
C
{\displaystyle \int \operatorname {arcoth} \,x\,dx=x\,\operatorname {arcoth} \,x+{\frac {1}{2}}\ln {(x^{2}-1)}+C}
sửa
∫
cos
a
x
e
b
x
d
x
=
e
b
x
a
2
+
b
2
(
a
sin
a
x
+
b
cos
a
x
)
+
C
{\displaystyle \int \cos ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(a\sin ax+b\cos ax\right)+C}
∫
sin
a
x
e
b
x
d
x
=
e
b
x
a
2
+
b
2
(
b
sin
a
x
−
a
cos
a
x
)
+
C
{\displaystyle \int \sin ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(b\sin ax-a\cos ax\right)+C}
∫
cos
a
x
cosh
b
x
d
x
=
1
a
2
+
b
2
(
a
sin
a
x
cosh
b
x
+
b
cos
a
x
sinh
b
x
)
+
C
{\displaystyle \int \cos ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(a\sin ax\,\cosh bx+b\cos ax\,\sinh bx\right)+C}
∫
sin
a
x
cosh
b
x
d
x
=
1
a
2
+
b
2
(
b
sin
a
x
sinh
b
x
−
a
cos
a
x
cosh
b
x
)
+
C
{\displaystyle \int \sin ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(b\sin ax\,\sinh bx-a\cos ax\,\cosh bx\right)+C}
sửa
∫
|
(
a
x
+
b
)
n
|
d
x
=
(
a
x
+
b
)
n
+
2
a
(
n
+
1
)
|
a
x
+
b
|
+
C
[
n
is odd, and
n
≠
−
1
]
{\displaystyle \int \left|(ax+b)^{n}\right|\,dx={(ax+b)^{n+2} \over a(n+1)\left|ax+b\right|}+C\,\,[\,n{\text{ is odd, and }}n\neq -1\,]}
∫
|
sin
a
x
|
d
x
=
−
1
a
|
sin
a
x
|
cot
a
x
+
C
{\displaystyle \int \left|\sin {ax}\right|\,dx={-1 \over a}\left|\sin {ax}\right|\cot {ax}+C}
∫
|
cos
a
x
|
d
x
=
1
a
|
cos
a
x
|
tan
a
x
+
C
{\displaystyle \int \left|\cos {ax}\right|\,dx={1 \over a}\left|\cos {ax}\right|\tan {ax}+C}
∫
|
tan
a
x
|
d
x
=
tan
(
a
x
)
[
−
ln
|
cos
a
x
|
]
a
|
tan
a
x
|
+
C
{\displaystyle \int \left|\tan {ax}\right|\,dx={\tan(ax)[-\ln \left|\cos {ax}\right|] \over a\left|\tan {ax}\right|}+C}
∫
|
csc
a
x
|
d
x
=
−
ln
|
csc
a
x
+
cot
a
x
|
sin
a
x
a
|
sin
a
x
|
+
C
{\displaystyle \int \left|\csc {ax}\right|\,dx={-\ln \left|\csc {ax}+\cot {ax}\right|\sin {ax} \over a\left|\sin {ax}\right|}+C}
∫
|
sec
a
x
|
d
x
=
ln
|
sec
a
x
+
tan
a
x
|
cos
a
x
a
|
cos
a
x
|
+
C
{\displaystyle \int \left|\sec {ax}\right|\,dx={\ln \left|\sec {ax}+\tan {ax}\right|\cos {ax} \over a\left|\cos {ax}\right|}+C}
∫
|
cot
a
x
|
d
x
=
tan
(
a
x
)
[
ln
|
sin
a
x
|
]
a
|
tan
a
x
|
+
C
{\displaystyle \int \left|\cot {ax}\right|\,dx={\tan(ax)[\ln \left|\sin {ax}\right|] \over a\left|\tan {ax}\right|}+C}
sửa
∫
Ci
(
x
)
d
x
=
x
Ci
(
x
)
−
sin
x
{\displaystyle \int \operatorname {Ci} (x)dx=x\,\operatorname {Ci} (x)-\sin x}
∫
Si
(
x
)
d
x
=
x
Si
(
x
)
+
cos
x
{\displaystyle \int \operatorname {Si} (x)dx=x\,\operatorname {Si} (x)+\cos x}
∫
Ei
(
x
)
d
x
=
x
Ei
(
x
)
−
e
x
{\displaystyle \int \operatorname {Ei} (x)dx=x\,\operatorname {Ei} (x)-e^{x}}
∫
li
(
x
)
d
x
=
x
li
(
x
)
−
Ei
(
2
ln
x
)
{\displaystyle \int \operatorname {li} (x)dx=x\,\operatorname {li} (x)-\operatorname {Ei} (2\ln x)}
∫
li
(
x
)
x
d
x
=
ln
x
li
(
x
)
−
x
{\displaystyle \int {\frac {\operatorname {li} (x)}{x}}\,dx=\ln x\,\operatorname {li} (x)-x}
∫
erf
(
x
)
d
x
=
e
−
x
2
π
+
x
erf
(
x
)
{\displaystyle \int \operatorname {erf} (x)\,dx={\frac {e^{-x^{2}}}{\sqrt {\pi }}}+x\,{\text{erf}}(x)}
![{\displaystyle \int \operatorname {arcsch} \,x\,dx=x\,\operatorname {arcsch} \,x+\ln {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca1160efc8b0a76d64c89bca2625a9c7828dd9f)
![{\displaystyle \int \left|(ax+b)^{n}\right|\,dx={(ax+b)^{n+2} \over a(n+1)\left|ax+b\right|}+C\,\,[\,n{\text{ is odd, and }}n\neq -1\,]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9db551fb843fee76fb6a9325999ab8c0515a8d1f)
![{\displaystyle \int \left|\tan {ax}\right|\,dx={\tan(ax)[-\ln \left|\cos {ax}\right|] \over a\left|\tan {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/138aa84438bbb1353aa6d046f7987681f7223958)
![{\displaystyle \int \left|\cot {ax}\right|\,dx={\tan(ax)[\ln \left|\sin {ax}\right|] \over a\left|\tan {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8bc94eb6f71b1b8b2f126a0dc1182e080ee756)
