Integreren is een basisbewerking uit de analyse . Aangezien integreren niet, zoals bij differentiëren , door eenvoudige regels plaatsvindt, zijn tabellen met veel voorkomende integralen een handig hulpmiddel. In de onderstaande lijst van integralen wordt van een groot aantal verschillende functies de primitieve functie gegeven.
Er zijn lijsten van integralen:
Er wordt van alle genoemde integralen de primitieve functie gegeven. De primitieve functie van een functie is tot op de integratieconstante na bepaald. De integratieconstante
C
{\displaystyle C}
beginvoorwaarde of randvoorwaarde worden bepaald. Bij de hier gegeven integralen worden de onder- en de bovengrens van het interval , waarover wordt geïntegreerd, niet gegeven. Oneigenlijke integralen worden apart behandeld.
Rekenregels bij het integreren
bewerken
∫
c
f
(
x
)
d
x
=
c
∫
f
(
x
)
d
x
{\displaystyle \int cf(x)\ {\rm {d}}x=c\int f(x)\ {\rm {dx}}}
∫
(
f
(
x
)
+
g
(
x
)
)
d
x
=
∫
f
(
x
)
d
x
+
∫
g
(
x
)
d
x
{\displaystyle \int (f(x)+g(x))\ {\rm {d}}x=\int f(x)\ {\rm {d}}x+\int g(x)\ {\rm {d}}x}
∫
f
(
x
)
g
′
(
x
)
d
x
=
f
(
x
)
g
(
x
)
−
∫
f
′
(
x
)
g
(
x
)
d
x
{\displaystyle \int f(x)g'(x)\ {\rm {d}}x=f(x)g(x)-\int f'(x)g(x)\ {\rm {d}}x}
∫
a
b
d
F
(
x
)
d
x
d
x
=
[
F
(
x
)
]
a
b
=
F
(
b
)
−
F
(
a
)
{\displaystyle \int _{a}^{b}{\frac {{\rm {d}}F(x)}{{\rm {d}}x}}\ {\rm {d}}x=[F(x)]_{a}^{b}=F(b)-F(a)}
Meervoudige integraal als herhaalde integraal
∬
f
(
x
,
y
)
d
x
d
y
=
∫
(
∫
f
(
x
,
y
)
d
x
)
d
y
{\displaystyle \iint f(x,y)\ {\rm {d}}x{\rm {d}}y=\int \left(\int f(x,y)\ {\rm {d}}x\right){\rm {d}}y}
∫
f
(
g
(
t
)
)
g
′
(
t
)
d
t
=
∫
f
(
x
)
d
x
{\displaystyle \int f(g(t))\ g'(t)\ {\rm {d}}t=\int f(x)\ {\rm {d}}x}
Integralen van standaardfuncties
bewerken
∫
1
d
x
=
x
+
C
{\displaystyle \int 1\ {\rm {d}}x=x+C}
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
C
als
n
≠
−
1
{\displaystyle \int x^{n}\ {\rm {d}}x={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ als }}n\neq -1}
∫
1
x
d
x
=
ln
|
x
|
+
C
{\displaystyle \int {\frac {1}{x}}\ {\rm {d}}x=\ln {\left|x\right|}+C}
∫
1
a
2
+
x
2
d
x
=
1
a
arctan
x
a
+
C
{\displaystyle \int {\frac {1}{a^{2}+x^{2}}}\ {\rm {d}}x={\frac {1}{a}}\arctan {\frac {x}{a}}+C}
∫
1
x
(
a
+
b
x
)
d
x
=
1
a
ln
|
x
a
+
b
x
|
+
C
{\displaystyle \int {\frac {1}{x\left(a+bx\right)}}\ {\rm {d}}x={\frac {1}{a}}\ln \left|{\frac {x}{a+bx}}\right|+C}
∫
1
a
x
2
+
b
x
+
c
d
x
=
{
1
b
2
−
4
a
c
ln
|
2
a
x
+
b
−
b
2
−
4
a
c
2
a
x
+
b
+
b
2
−
4
a
c
|
+
C
als
b
2
>
4
a
c
2
4
a
c
−
b
2
arctan
2
a
x
+
b
4
a
c
−
b
2
+
C
als
b
2
<
4
a
c
{\displaystyle \int {\frac {1}{ax^{2}+bx+c}}\ {\rm {d}}x=\left\{{\begin{matrix}{\cfrac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\cfrac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|+C&{\mbox{als}}\ b^{2}>4ac\\{\cfrac {2}{\sqrt {4ac-b^{2}}}}\arctan {\cfrac {2ax+b}{\sqrt {4ac-b^{2}}}}+C&{\mbox{als}}\ b^{2}<4ac\end{matrix}}\right.}
∫
x
a
x
2
+
b
x
+
c
d
x
=
1
2
a
ln
|
a
x
2
+
b
x
+
c
|
−
b
2
a
∫
1
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\frac {x}{ax^{2}+bx+c}}\ {\rm {d}}x={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {1}{ax^{2}+bx+c}}\ {\rm {d}}x}
Irrationale functies
bewerken
∫
d
u
a
2
−
u
2
=
arcsin
u
a
+
C
{\displaystyle \int {{\rm {d}}u \over {\sqrt {a^{2}-u^{2}}}}=\arcsin {u \over a}+C}
∫
−
d
u
a
2
−
u
2
=
arccos
u
a
+
C
{\displaystyle \int {-{\rm {d}}u \over {\sqrt {a^{2}-u^{2}}}}=\arccos {u \over a}+C}
∫
d
u
u
u
2
−
a
2
=
1
a
arcsec
|
u
|
a
+
C
{\displaystyle \int {{\rm {d}}u \over u{\sqrt {u^{2}-a^{2}}}}={1 \over a}{\mbox{arcsec}}\ {|u| \over a}+C}
∫
a
2
−
x
2
d
x
=
x
2
a
2
−
x
2
+
a
2
2
arcsin
x
a
+
C
,
a
>
0
{\displaystyle \int {\sqrt {a^{2}-x^{2}}}\ {\rm {d}}x={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+C,\quad a>0}
∫
(
a
2
−
x
2
)
3
2
d
x
=
x
8
(
5
a
2
−
2
x
2
)
a
2
−
x
2
+
3
a
4
8
arcsin
x
a
+
C
,
a
>
0
{\displaystyle \int \left(a^{2}-x^{2}\right)^{\frac {3}{2}}\ {\rm {d}}x={\frac {x}{8}}\left(5a^{2}-2x^{2}\right){\sqrt {a^{2}-x^{2}}}+{\frac {3a^{4}}{8}}\arcsin {\frac {x}{a}}+C,\quad a>0}
∫
1
(
a
2
−
x
2
)
3
2
d
x
=
x
a
2
a
2
−
x
2
+
C
{\displaystyle \int {\frac {1}{\left(a^{2}-x^{2}\right)^{\frac {3}{2}}}}\ {\rm {d}}x={\frac {x}{a^{2}{\sqrt {a^{2}-x^{2}}}}}+C}
∫
x
a
+
b
x
d
x
=
2
(
3
b
x
−
2
a
)
(
a
+
b
x
)
3
2
15
b
2
+
C
{\displaystyle \int x{\sqrt {a+bx}}\ {\rm {d}}x={\frac {2\left(3bx-2a\right)\left(a+bx\right)^{\frac {3}{2}}}{15b^{2}}}+C}
∫
a
+
b
x
x
d
x
=
2
a
+
b
x
+
a
∫
1
x
a
+
b
x
d
x
{\displaystyle \int {\frac {\sqrt {a+bx}}{x}}\ {\rm {d}}x=2{\sqrt {a+bx}}+a\int {\frac {1}{x{\sqrt {a+bx}}}}\ {\rm {d}}x}
∫
x
a
+
b
x
d
x
=
2
(
b
x
−
2
a
)
a
+
b
x
3
b
2
+
C
{\displaystyle \int {\frac {x}{\sqrt {a+bx}}}\ {\rm {d}}x={\frac {2\left(bx-2a\right){\sqrt {a+bx}}}{3b^{2}}}+C}
∫
1
x
a
+
b
x
d
x
=
1
a
ln
|
a
+
b
x
−
a
a
+
b
x
+
a
|
+
C
,
a
>
0
{\displaystyle \int {\frac {1}{x{\sqrt {a+bx}}}}\ {\rm {d}}x={\frac {1}{\sqrt {a}}}\ln \left|{\frac {{\sqrt {a+bx}}-{\sqrt {a}}}{{\sqrt {a+bx}}+{\sqrt {a}}}}\right|+C,\quad a>0}
∫
1
x
a
+
b
x
d
x
=
2
−
a
arctan
|
a
+
b
x
−
a
|
+
C
,
a
<
0
{\displaystyle \int {\frac {1}{x{\sqrt {a+bx}}}}\ {\rm {d}}x={\frac {2}{\sqrt {-a}}}\arctan \left|{\sqrt {\frac {a+bx}{-a}}}\right|+C,\quad a<0}
∫
a
2
−
x
2
x
d
x
=
a
2
−
x
2
−
a
ln
|
a
+
a
2
+
x
2
x
|
+
C
{\displaystyle \int {\frac {\sqrt {a^{2}-x^{2}}}{x}}\ {\rm {d}}x={\sqrt {a^{2}-x^{2}}}-a\ln \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|+C}
∫
x
a
2
−
x
2
d
x
=
−
1
3
(
a
2
−
x
2
)
3
2
+
C
{\displaystyle \int x{\sqrt {a^{2}-x^{2}}}\ {\rm {d}}x=-{\frac {1}{3}}\left(a^{2}-x^{2}\right)^{\frac {3}{2}}+C}
∫
x
2
a
2
−
x
2
d
x
=
x
8
(
2
x
2
−
a
2
)
a
2
−
x
2
+
a
4
8
arcsin
x
a
+
C
,
a
>
0
{\displaystyle \int x^{2}{\sqrt {a^{2}-x^{2}}}\ {\rm {d}}x={\frac {x}{8}}\left(2x^{2}-a^{2}\right){\sqrt {a^{2}-x^{2}}}+{\frac {a^{4}}{8}}\arcsin {\frac {x}{a}}+C,\quad a>0}
∫
1
x
a
2
−
x
2
d
x
=
−
1
a
ln
|
a
+
a
2
−
x
2
x
|
+
C
{\displaystyle \int {\frac {1}{x{\sqrt {a^{2}-x^{2}}}}}\ {\rm {d}}x=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|+C}
∫
x
a
2
−
x
2
d
x
=
−
a
2
−
x
2
+
C
{\displaystyle \int {\frac {x}{\sqrt {a^{2}-x^{2}}}}\ {\rm {d}}x=-{\sqrt {a^{2}-x^{2}}}+C}
∫
x
2
a
2
−
x
2
d
x
=
−
x
2
a
2
−
x
2
+
a
2
2
arcsin
x
a
+
C
,
a
>
0
{\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}-x^{2}}}}\ {\rm {d}}x=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+C,\quad a>0}
∫
x
2
+
a
2
x
d
x
=
x
2
+
a
2
−
a
ln
|
a
+
x
2
+
a
2
x
|
+
C
{\displaystyle \int {\frac {\sqrt {x^{2}+a^{2}}}{x}}\ {\rm {d}}x={\sqrt {x^{2}+a^{2}}}-a\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|+C}
∫
x
2
−
a
2
x
d
x
=
x
2
−
a
2
−
a
arccos
a
|
x
|
+
C
,
a
>
0
{\displaystyle \int {\frac {\sqrt {x^{2}-a^{2}}}{x}}\ {\rm {d}}x={\sqrt {x^{2}-a^{2}}}-a\arccos {\frac {a}{|x|}}+C,\quad a>0}
∫
x
2
x
2
+
a
2
d
x
=
x
x
2
+
a
2
2
−
a
2
2
ln
(
x
+
x
2
+
a
2
)
+
C
{\displaystyle \int {\frac {x^{2}}{\sqrt {x^{2}+a^{2}}}}\ {\rm {d}}x={\frac {x{\sqrt {x^{2}+a^{2}}}}{2}}-{\frac {a^{2}}{2}}\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)+C}
∫
1
x
x
2
+
a
2
d
x
=
1
a
ln
|
x
a
+
x
2
+
a
2
|
+
C
{\displaystyle \int {\frac {1}{x{\sqrt {x^{2}+a^{2}}}}}\ {\rm {d}}x={\frac {1}{a}}\ln \left|{\frac {x}{a+{\sqrt {x^{2}+a^{2}}}}}\right|+C}
∫
1
x
2
x
2
±
a
2
d
x
=
∓
x
2
±
a
2
a
2
x
+
C
{\displaystyle \int {\frac {1}{x^{2}{\sqrt {x^{2}\pm a^{2}}}}}\ {\rm {d}}x=\mp {\frac {\sqrt {x^{2}\pm a^{2}}}{a^{2}x}}+C}
∫
1
x
2
±
a
2
d
x
=
ln
|
x
+
x
2
±
a
2
a
|
+
C
=
arcsinh
x
a
+
C
{\displaystyle \int {\frac {1}{\sqrt {x^{2}\pm a^{2}}}}\ {\rm {d}}x=\ln \left|{\frac {x+{\sqrt {x^{2}\pm a^{2}}}}{a}}\right|+C=\operatorname {arcsinh} {\frac {x}{a}}+C}
∫
1
a
x
2
+
b
x
+
c
d
x
=
1
a
ln
|
2
a
x
+
b
+
2
a
a
x
2
+
b
x
+
c
|
+
C
,
a
>
0
{\displaystyle \int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\ {\rm {d}}x={\frac {1}{\sqrt {a}}}\ln \left|2ax+b+2{\sqrt {a}}{\sqrt {ax^{2}+bx+c}}\right|+C,\quad a>0}
∫
1
a
x
2
+
b
x
+
c
d
x
=
1
−
a
arcsin
−
2
a
x
−
b
b
2
−
4
a
c
+
C
,
a
<
0
{\displaystyle \int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\ {\rm {d}}x={\frac {1}{\sqrt {-a}}}\arcsin {\frac {-2ax-b}{\sqrt {b^{2}-4ac}}}+C,\quad a<0}
∫
a
x
2
+
b
x
+
c
d
x
=
2
a
x
+
b
4
a
a
x
2
+
b
x
+
c
+
4
a
c
−
b
2
8
a
∫
1
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\sqrt {ax^{2}+bx+c}}\ {\rm {d}}x={\frac {2ax+b}{4a}}{\sqrt {ax^{2}+bx+c}}+{\frac {4ac-b^{2}}{8a}}\int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\ {\rm {d}}x}
∫
x
a
x
2
+
b
x
+
c
d
x
=
a
x
2
+
b
x
+
c
a
−
b
2
a
∫
1
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\frac {x}{\sqrt {ax^{2}+bx+c}}}\ {\rm {d}}x={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\ {\rm {d}}x}
∫
1
x
a
x
2
+
b
x
+
c
d
x
=
−
1
c
ln
|
2
c
a
x
2
+
b
x
+
c
+
b
x
+
2
c
x
|
+
C
,
c
>
0
{\displaystyle \int {\frac {1}{x{\sqrt {ax^{2}+bx+c}}}}\ {\rm {d}}x={\frac {-1}{\sqrt {c}}}\ln \left|{\frac {2{\sqrt {c}}{\sqrt {ax^{2}+bx+c}}+bx+2c}{x}}\right|+C,\quad c>0}
∫
1
x
a
x
2
+
b
x
+
c
d
x
=
1
−
c
arcsin
b
x
+
2
c
|
x
|
b
2
−
4
a
c
+
C
,
c
<
0
{\displaystyle \int {\frac {1}{x{\sqrt {ax^{2}+bx+c}}}}\ {\rm {d}}x={\frac {1}{\sqrt {-c}}}\arcsin {\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac}}}}+C,\quad c<0}
∫
x
3
x
2
+
a
2
d
x
=
(
1
5
x
2
−
2
15
a
2
)
(
x
2
+
a
2
)
3
+
C
{\displaystyle \int x^{3}{\sqrt {x^{2}+a^{2}}}\ {\rm {d}}x=\left({\frac {1}{5}}x^{2}-{\frac {2}{15}}a^{2}\right){\sqrt {\left(x^{2}+a^{2}\right)^{3}}}+C}
∫
x
2
±
a
2
x
4
d
x
=
∓
(
x
2
+
a
2
)
3
3
a
2
x
3
+
C
{\displaystyle \int {\frac {\sqrt {x^{2}\pm a^{2}}}{x^{4}}}\ {\rm {d}}x={\frac {\mp {\sqrt {\left(x^{2}+a^{2}\right)^{3}}}}{3a^{2}x^{3}}}+C}
∫
e
x
d
x
=
e
x
+
C
{\displaystyle \int e^{x}\ {\rm {d}}x=e^{x}+C}
∫
a
x
d
x
=
a
x
ln
a
+
C
{\displaystyle \int a^{x}\ {\rm {d}}x={\frac {a^{x}}{\ln {a}}}+C}
∫
e
a
x
d
x
=
e
a
x
a
+
C
{\displaystyle \int e^{ax}\ {\rm {d}}x={\frac {e^{ax}}{a}}+C}
∫
a
b
x
d
x
=
a
b
x
b
ln
a
+
C
{\displaystyle \int a^{bx}\ {\rm {d}}x={\frac {a^{bx}}{b\ln {a}}}+C}
∫
x
n
e
a
x
d
x
=
x
n
e
a
x
a
−
n
a
∫
x
n
−
1
e
a
x
d
x
{\displaystyle \int x^{n}e^{ax}\ {\rm {d}}x={\frac {x^{n}e^{ax}}{a}}-{\frac {n}{a}}\int x^{n-1}e^{ax}\ {\rm {d}}x}
∫
ln
x
d
x
=
x
ln
x
−
x
+
C
{\displaystyle \int \ln {x}\ {\rm {d}}x=x\ln {x}-x+C}
∫
log
b
x
d
x
=
x
log
b
x
−
x
log
b
e
+
C
{\displaystyle \int \log _{b}{x}\ {\rm {d}}x=x\log _{b}{x}-x\log _{b}{e}+C}
∫
x
n
ln
a
x
d
x
=
x
n
+
1
(
ln
a
x
n
+
1
−
1
(
n
+
1
)
2
)
+
C
{\displaystyle \int x^{n}\ln ax\ {\rm {d}}x=x^{n+1}\left({\frac {\ln ax}{n+1}}-{\frac {1}{(n+1)^{2}}}\right)+C}
∫
x
n
(
ln
a
x
)
m
d
x
=
x
n
+
1
n
+
1
(
ln
a
x
)
m
−
m
n
+
1
∫
x
n
(
ln
a
x
)
m
−
1
d
x
{\displaystyle \int x^{n}\left(\ln ax\right)^{m}\ {\rm {d}}x={\frac {x^{n+1}}{n+1}}\left(\ln ax\right)^{m}-{\frac {m}{n+1}}\int x^{n}\left(\ln ax\right)^{m-1}\ {\rm {d}}x}
∫
sin
x
d
x
=
−
cos
x
+
C
{\displaystyle \int \sin {x}\ {\rm {d}}x=-\cos {x}+C}
∫
cos
x
d
x
=
sin
x
+
C
{\displaystyle \int \cos {x}\ {\rm {d}}x=\sin {x}+C}
∫
tan
x
d
x
=
−
ln
|
cos
x
|
+
C
{\displaystyle \int \tan {x}\ {\rm {d}}x=-\ln {\left|\cos {x}\right|}+C}
∫
cot
x
d
x
=
ln
|
sin
x
|
+
C
{\displaystyle \int \cot {x}\ {\rm {d}}x=\ln {\left|\sin {x}\right|}+C}
∫
sec
x
d
x
=
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \int \sec {x}\ {\rm {d}}x=\ln {\left|\sec {x}+\tan {x}\right|}+C}
∫
csc
x
d
x
=
ln
|
csc
x
−
cot
x
|
+
C
{\displaystyle \int \csc {x}\ {\rm {d}}x=\ln {\left|\csc {x}-\cot {x}\right|}+C}
∫
1
sin
x
d
x
=
ln
|
tan
1
2
x
|
+
C
=
ln
|
1
sin
x
−
cot
x
|
+
C
{\displaystyle \int {\frac {1}{\sin x}}\ {\rm {d}}x=\ln \left|\tan {\tfrac {1}{2}}x\right|+C=\ln \left|{\frac {1}{\sin x}}-\cot x\right|+C}
∫
1
cos
x
d
x
=
ln
|
tan
1
2
x
+
1
4
π
|
+
C
=
ln
|
1
cos
x
+
tan
x
|
+
C
{\displaystyle \int {\frac {1}{\cos x}}\ {\rm {d}}x=\ln \left|\tan {\tfrac {1}{2}}x+{\tfrac {1}{4}}\pi \right|+C=\ln \left|{\frac {1}{\cos x}}+\tan x\right|+C}
∫
arcsin
x
a
d
x
=
x
arcsin
x
a
+
a
2
−
x
2
+
C
,
a
>
0
{\displaystyle \int \arcsin {\frac {x}{a}}\ {\rm {d}}x=x\arcsin {\frac {x}{a}}+{\sqrt {a^{2}-x^{2}}}+C,\quad a>0}
∫
arccos
x
a
d
x
=
x
arccos
x
a
−
a
2
−
x
2
+
C
,
a
>
0
{\displaystyle \int \arccos {\frac {x}{a}}\ {\rm {d}}x=x\arccos {\frac {x}{a}}-{\sqrt {a^{2}-x^{2}}}+C,\quad a>0}
∫
arctan
x
a
d
x
=
x
arctan
x
a
−
a
2
ln
(
a
2
+
x
2
)
+
C
,
a
>
0
{\displaystyle \int \arctan {\frac {x}{a}}\ {\rm {d}}x=x\arctan {\frac {x}{a}}-{\frac {a}{2}}\ln \left(a^{2}+x^{2}\right)+C,\quad a>0}
∫
1
cos
2
x
d
x
=
∫
sec
2
x
d
x
=
tan
x
+
C
{\displaystyle \int {\frac {1}{\cos ^{2}x}}\ {\rm {d}}x=\int \sec ^{2}x\ {\rm {d}}x=\tan x+C}
∫
1
sin
2
x
d
x
=
∫
csc
2
x
d
x
=
−
cot
x
+
C
{\displaystyle \int {\frac {1}{\sin ^{2}x}}\ {\rm {d}}x=\int \csc ^{2}x\ {\rm {d}}x=-\cot x+C}
∫
sec
x
tan
x
d
x
=
sec
x
+
C
{\displaystyle \int \sec {x}\ \tan {x}\ {\rm {d}}x=\sec {x}+C}
∫
csc
x
cot
x
d
x
=
−
csc
x
+
C
{\displaystyle \int \csc {x}\ \cot {x}\ {\rm {d}}x=-\csc {x}+C}
∫
sin
2
x
d
x
=
1
2
(
x
−
sin
x
cos
x
)
+
C
{\displaystyle \int \sin ^{2}x\ {\rm {d}}x={\tfrac {1}{2}}(x-\sin x\cos x)+C}
∫
cos
2
x
d
x
=
1
2
(
x
+
sin
x
cos
x
)
+
C
{\displaystyle \int \cos ^{2}x\ {\rm {d}}x={\tfrac {1}{2}}(x+\sin x\cos x)+C}
∫
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\ {\rm {d}}x=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\ {\rm {d}}x}
∫
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\ {\rm {d}}x={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\ {\rm {d}}x}
∫
tan
n
x
d
x
=
tan
n
−
1
x
n
−
1
−
∫
tan
n
−
2
x
d
x
,
n
≠
1
{\displaystyle \int \tan ^{n}x\ {\rm {d}}x={\frac {\tan ^{n-1}x}{n-1}}-\int \tan ^{n-2}x\ {\rm {d}}x,\quad n\neq 1}
∫
cot
n
x
d
x
=
−
cot
n
−
1
x
n
−
1
−
∫
cot
n
−
2
x
d
x
,
n
≠
1
{\displaystyle \int \cot ^{n}x\ {\rm {d}}x=-{\frac {\cot ^{n-1}x}{n-1}}-\int \cot ^{n-2}x\ {\rm {d}}x,\quad n\neq 1}
∫
sec
n
x
d
x
=
tan
x
sec
n
−
2
x
n
−
1
+
n
−
2
n
−
1
∫
sec
n
−
2
x
d
x
,
n
≠
1
{\displaystyle \int \sec ^{n}x\ {\rm {d}}x={\frac {\tan x\sec ^{n-2}x}{n-1}}+{\frac {n-2}{n-1}}\int \sec ^{n-2}x\ {\rm {d}}x,\quad n\neq 1}
∫
csc
n
x
d
x
=
−
cot
x
csc
n
−
2
x
n
−
1
+
n
−
2
n
−
1
∫
csc
n
−
2
x
d
x
,
n
≠
1
{\displaystyle \int \csc ^{n}x\ {\rm {d}}x=-{\frac {\cot x\csc ^{n-2}x}{n-1}}+{\frac {n-2}{n-1}}\int \csc ^{n-2}x\ {\rm {d}}x,\quad n\neq 1}
∫
sin
a
x
sin
b
x
d
x
=
sin
(
a
−
b
)
x
2
(
a
−
b
)
−
sin
(
a
+
b
)
x
2
(
a
+
b
)
+
C
,
a
2
≠
b
2
{\displaystyle \int \sin ax\sin bx\ {\rm {d}}x={\frac {\sin(a-b)x}{2(a-b)}}-{\frac {\sin(a+b)x}{2(a+b)}}+C,\quad a^{2}\neq b^{2}}
∫
sin
a
x
cos
b
x
d
x
=
−
cos
(
a
−
b
)
x
2
(
a
−
b
)
−
cos
(
a
+
b
)
x
2
(
a
+
b
)
+
C
,
a
2
≠
b
2
{\displaystyle \int \sin ax\cos bx\ {\rm {d}}x=-{\frac {\cos(a-b)x}{2(a-b)}}-{\frac {\cos(a+b)x}{2(a+b)}}+C,\quad a^{2}\neq b^{2}}
∫
cos
a
x
cos
b
x
d
x
=
sin
(
a
−
b
)
x
2
(
a
−
b
)
+
sin
(
a
+
b
)
x
2
(
a
+
b
)
+
C
,
a
2
≠
b
2
{\displaystyle \int \cos ax\cos bx\ {\rm {d}}x={\frac {\sin(a-b)x}{2(a-b)}}+{\frac {\sin(a+b)x}{2(a+b)}}+C,\quad a^{2}\neq b^{2}}
∫
sec
x
tan
x
d
x
=
sec
x
+
C
{\displaystyle \int \sec x\tan x\ {\rm {d}}x=\sec x+C}
∫
csc
x
cot
x
d
x
=
−
csc
x
+
C
{\displaystyle \int \csc x\cot x\ {\rm {d}}x=-\csc x+C}
∫
cos
m
x
sin
n
x
d
x
=
cos
m
−
1
x
sin
n
+
1
x
m
+
n
+
m
−
1
m
+
n
∫
cos
m
−
2
x
sin
n
x
d
x
{\displaystyle \int \cos ^{m}x\sin ^{n}x\ {\rm {d}}x={\frac {\cos ^{m-1}x\sin ^{n+1}x}{m+n}}+{\frac {m-1}{m+n}}\int \cos ^{m-2}x\sin ^{n}x\ {\rm {d}}x}
=
−
sin
n
−
1
x
cos
m
+
1
x
m
+
n
+
n
−
1
m
+
n
∫
cos
m
x
sin
n
−
2
x
d
x
{\displaystyle =-{\frac {\sin ^{n-1}x\cos ^{m+1}x}{m+n}}+{\frac {n-1}{m+n}}\int \cos ^{m}x\sin ^{n-2}x\ {\rm {d}}x}
∫
x
n
sin
a
x
d
x
=
−
1
a
x
n
cos
a
x
+
n
a
∫
x
n
−
1
cos
a
x
d
x
{\displaystyle \int x^{n}\sin ax\ {\rm {d}}x=-{\frac {1}{a}}x^{n}\cos ax+{\frac {n}{a}}\int x^{n-1}\cos ax\ {\rm {d}}x}
∫
x
n
cos
a
x
d
x
=
1
a
x
n
sin
a
x
−
n
a
∫
x
n
−
1
sin
a
x
d
x
{\displaystyle \int x^{n}\cos ax\ {\rm {d}}x={\frac {1}{a}}x^{n}\sin ax-{\frac {n}{a}}\int x^{n-1}\sin ax\ {\rm {d}}x}
∫
e
a
x
sin
b
x
d
x
=
e
a
x
(
a
sin
b
x
−
b
cos
b
x
)
a
2
+
b
2
+
C
{\displaystyle \int e^{ax}\sin bx\ {\rm {d}}x={\frac {e^{ax}\left(a\sin bx-b\cos bx\right)}{a^{2}+b^{2}}}+C}
∫
e
a
x
cos
b
x
d
x
=
e
a
x
(
b
sin
b
x
+
a
cos
b
x
)
a
2
+
b
2
+
C
{\displaystyle \int e^{ax}\cos bx\ {\rm {d}}x={\frac {e^{ax}\left(b\sin bx+a\cos bx\right)}{a^{2}+b^{2}}}+C}
∫
sinh
x
d
x
=
cosh
x
+
C
{\displaystyle \int \sinh x\ {\rm {d}}x=\cosh x+C}
∫
cosh
x
d
x
=
sinh
x
+
C
{\displaystyle \int \cosh x\ {\rm {d}}x=\sinh x+C}
∫
tanh
x
d
x
=
ln
|
cosh
x
|
+
C
{\displaystyle \int \tanh x\ {\rm {d}}x=\ln |\cosh x|+C}
∫
csch
x
d
x
=
ln
|
tanh
x
2
|
+
C
{\displaystyle \int \operatorname {csch} \ x\ {\rm {d}}x=\ln \left|\tanh {x \over 2}\right|+C}
∫
sech
x
d
x
=
arctan
(
sinh
x
)
+
C
{\displaystyle \int \operatorname {sech} \ x\ {\rm {d}}x=\arctan(\sinh x)+C}
∫
coth
x
d
x
=
ln
|
sinh
x
|
+
C
{\displaystyle \int \coth x\ {\rm {d}}x=\ln |\sinh x|+C}
∫
sinh
2
x
d
x
=
1
4
sinh
2
x
−
1
2
x
+
C
{\displaystyle \int \sinh ^{2}x\ {\rm {d}}x={\frac {1}{4}}\sinh 2x-{\frac {1}{2}}x+C}
∫
cosh
2
x
d
x
=
1
4
sinh
2
x
+
1
2
x
+
C
{\displaystyle \int \cosh ^{2}x\ {\rm {d}}x={\frac {1}{4}}\sinh 2x+{\frac {1}{2}}x+C}
∫
sech
2
x
d
x
=
tanh
x
+
C
{\displaystyle \int \operatorname {sech} ^{2}x\ {\rm {d}}x=\tanh x+C}
∫
1
sinh
2
x
d
x
=
−
coth
x
+
C
{\displaystyle \int {\frac {1}{\sinh ^{2}x}}\ {\rm {d}}x=-\operatorname {\coth } x+C}
∫
sinh
−
1
x
a
d
x
=
x
sinh
−
1
x
a
−
x
2
+
a
2
+
C
{\displaystyle \int \sinh ^{-1}{\frac {x}{a}}\ {\rm {d}}x=x\sinh ^{-1}{\frac {x}{a}}-{\sqrt {x^{2}+a^{2}}}+C}
∫
cosh
−
1
x
a
d
x
=
x
cosh
−
1
x
a
−
x
2
−
a
2
+
C
,
cosh
−
1
x
a
<
0
en
a
>
0
{\displaystyle \int \cosh ^{-1}{\frac {x}{a}}\ {\rm {d}}x=x\cosh ^{-1}{\frac {x}{a}}-{\sqrt {x^{2}-a^{2}}}+C,\quad \cosh ^{-1}{\frac {x}{a}}<0\ {\mbox{ en }}\ a>0}
∫
cosh
−
1
x
a
d
x
=
x
cosh
−
1
x
a
+
x
2
−
a
2
+
C
,
cosh
−
1
x
a
<
0
en
a
>
0
{\displaystyle \int \cosh ^{-1}{\frac {x}{a}}\ {\rm {d}}x=x\cosh ^{-1}{\frac {x}{a}}+{\sqrt {x^{2}-a^{2}}}+C,\quad \cosh ^{-1}{\frac {x}{a}}<0\ {\mbox{ en }}\ a>0}
∫
tanh
−
1
x
a
d
x
=
x
tanh
−
1
x
a
+
a
2
ln
|
a
2
−
x
2
|
+
C
{\displaystyle \int \tanh ^{-1}{\frac {x}{a}}\ {\rm {d}}x=x\tanh ^{-1}{\frac {x}{a}}+{\frac {a}{2}}\ln \left|a^{2}-x^{2}\right|+C}
∫
sech
x
tanh
x
d
x
=
−
sech
x
+
C
{\displaystyle \int \operatorname {sech} x\tanh x\ {\rm {d}}x=-\operatorname {sech} x+C}
∫
csch
x
coth
x
d
x
=
−
csch
x
+
C
{\displaystyle \int \operatorname {csch} x\coth x\ {\rm {d}}x=-\operatorname {csch} x+C}
![{\displaystyle \int _{a}^{b}{\frac {{\rm {d}}F(x)}{{\rm {d}}x}}\ {\rm {d}}x=[F(x)]_{a}^{b}=F(b)-F(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b010650e0b38aea3a5190db2e36ddc732fdf28e7)
