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Toto je seznam základních integrálů (primitivních funkcí) často používaných ve výuce a v praxi. Odvození obvykle probíhá tak, že se derivuje primitivní funkce.
∫
0
d
x
=
c
{\displaystyle \int {0}\,\mathrm {d} x=c}
∫
a
d
x
=
a
x
+
c
{\displaystyle \int {a}\,\mathrm {d} x=ax+c}
∫
x
n
d
x
=
1
n
+
1
x
n
+
1
+
c
pro
x
>
0
,
n
∈
R
a
n
≠
−
1
{\displaystyle \int {x^{n}}\,\mathrm {d} x={\frac {1}{n+1}}x^{n+1}+c{\mbox{ pro }}x>0,n\in \mathbb {R} {\mbox{ a }}n\neq -1}
přirozená
n
{\displaystyle n}
x
{\displaystyle x}
∫
1
x
d
x
=
ln
|
x
|
+
c
pro
x
≠
0
{\displaystyle \int {\frac {1}{x}}\,\mathrm {d} x=\ln |x|+c{\mbox{ pro }}x\neq 0}
∫
e
x
d
x
=
e
x
+
c
{\displaystyle \int {\mathrm {e} ^{x}}\,\mathrm {d} x=\mathrm {e} ^{x}+c}
∫
a
x
d
x
=
a
x
ln
(
a
)
+
c
pro
a
>
0
,
a
≠
1
{\displaystyle \int {a^{x}}\,\mathrm {d} x={\frac {a^{x}}{\ln(a)}}\ +c{\mbox{ pro }}a>0,a\neq 1}
∫
sin
x
d
x
=
−
cos
x
+
c
{\displaystyle \int {\sin x}\,\mathrm {d} x=-\cos x+c}
∫
cos
x
d
x
=
sin
x
+
c
{\displaystyle \int {\cos x}\,\mathrm {d} x=\sin x+c}
∫
1
sin
2
x
d
x
=
−
cotg
x
+
c
pro
x
≠
n
π
{\displaystyle \int {\frac {1}{\sin ^{2}x}}\,\mathrm {d} x=-\operatorname {cotg} \,x+c{\mbox{ pro }}x\neq n\pi }
n
{\displaystyle n}
celé číslo .
∫
1
cos
2
x
d
x
=
tg
x
+
c
pro
x
≠
(
2
n
+
1
)
π
2
{\displaystyle \int {\frac {1}{\cos ^{2}x}}\,\mathrm {d} x=\operatorname {tg} \,x+c{\mbox{ pro }}x\neq (2n+1){\frac {\pi }{2}}}
n
{\displaystyle n}
celé číslo .
∫
1
1
+
x
2
d
x
=
arctg
x
+
c
1
=
−
arccotg
x
+
c
2
{\displaystyle \int {\frac {1}{1+x^{2}}}\mathrm {d} x=\operatorname {arctg} x+c_{1}=-\operatorname {arccotg} x+c_{2}}
∫
1
1
−
x
2
d
x
=
arcsin
x
+
c
1
=
−
arccos
x
+
c
2
pro
−
1
<
x
<
1
{\displaystyle \int {\frac {1}{\sqrt {1-x^{2}}}}\mathrm {d} x=\operatorname {arcsin} x+c_{1}=-\operatorname {arccos} x+c_{2}{\mbox{ pro }}-1<x<1}
∫
1
1
−
x
2
d
x
=
{
1
2
ln
|
1
+
x
1
−
x
|
+
c
,
pro
|
x
|
≠
1
arctgh
x
+
c
,
pro
|
x
|
<
1
arccotgh
x
+
c
pro
|
x
|
>
1
{\displaystyle \int {\frac {1}{1-x^{2}}}\mathrm {d} x=\left\{{\begin{matrix}{\frac {1}{2}}\ln {|{\frac {1+x}{1-x}}|}+c,&{\mbox{ pro }}|x|\neq 1\\\operatorname {arctgh} x+c,&{\mbox{ pro }}|x|<1\\\operatorname {arccotgh} x+c&{\mbox{ pro }}|x|>1\end{matrix}}\right.}
∫
sinh
x
d
x
=
cosh
x
+
c
{\displaystyle \int \sinh x\,\mathrm {d} x=\cosh x+c}
∫
cosh
x
d
x
=
sinh
x
+
c
{\displaystyle \int \cosh x\,\mathrm {d} x=\sinh x+c}
∫
1
sinh
2
x
d
x
=
−
cotgh
x
+
c
pro
x
≠
0
{\displaystyle \int {\frac {1}{\sinh ^{2}x}}\mathrm {d} x=-\operatorname {cotgh} x+c{\mbox{ pro }}x\neq 0}
∫
1
cosh
2
x
d
x
=
tgh
x
+
c
{\displaystyle \int {\frac {1}{\cosh ^{2}x}}\mathrm {d} x=\operatorname {tgh} x+c}
∫
1
x
2
+
1
d
x
=
ln
(
x
+
x
2
+
1
)
+
c
=
arcsinh
x
+
c
{\displaystyle \int {\frac {1}{\sqrt {x^{2}+1}}}\mathrm {d} x=\ln(x+{\sqrt {x^{2}+1}})+c=\operatorname {arcsinh} x+c}
∫
1
x
2
−
1
d
x
=
{
ln
|
x
+
x
2
−
1
|
+
c
,
pro
|
x
|
>
1
arcosh
x
+
c
,
pro
x
>
1
{\displaystyle \int {\frac {1}{\sqrt {x^{2}-1}}}\mathrm {d} x=\left\{{\begin{matrix}\ln {|x+{\sqrt {x^{2}-1}}|}+c,&{\mbox{ pro }}|x|>1\\\operatorname {arcosh} x+c,&{\mbox{ pro }}x>1\end{matrix}}\right.}
∫
[
f
(
x
)
±
g
(
x
)
]
d
x
=
∫
f
(
x
)
d
x
±
∫
g
(
x
)
d
x
{\displaystyle \int [f(x)\pm g(x)]\,\mathrm {d} x=\int f(x)\,\mathrm {d} x\pm \int g(x)\,\mathrm {d} x}
∫
k
f
(
x
)
d
x
=
k
∫
f
(
x
)
d
x
{\displaystyle \int k\,f(x)\,\mathrm {d} x=k\int f(x)\,\mathrm {d} x}
reálné číslo k 
![{\displaystyle \int [f(x)\pm g(x)]\,\mathrm {d} x=\int f(x)\,\mathrm {d} x\pm \int g(x)\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36e5d84ed1f8183fb1c94a0906d6b27870b43edc)
