Pagrindinis puslapis
Atsitiktinis
Prisijungti
Nustatymai
Aukoti
Apie Vikiknygas
Atsakomybės apribojimas
Paieška
Matematika/Integralų lentelė
Kalba
Stebėti
Keisti
<
Matematika
Pagrindiniai ir dažniausiai pasitaikantys
integralai
:
∫
0
d
x
=
C
{\displaystyle \int 0\;{\mathsf {d}}x=C}
∫
a
d
x
=
a
x
+
C
{\displaystyle \int a\;{\mathsf {d}}x=ax+C}
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
C
{\displaystyle \int x^{n}\;{\mathsf {d}}x={\frac {x^{n+1}}{n+1}}+C}
∫
d
x
x
=
ln
|
x
|
+
C
{\displaystyle \int {\frac {{\mathsf {d}}x}{x}}=\ln \left|x\right|+C}
∫
e
x
d
x
=
e
x
+
C
{\displaystyle \int {\mathsf {e}}^{x}\;{\mathsf {d}}x={\mathsf {e}}^{x}+C}
∫
a
x
d
x
=
a
x
ln
a
+
C
{\displaystyle \int a^{x}\;{\mathsf {d}}x={\frac {a^{x}}{\ln a}}+C}
∫
d
x
x
2
+
1
=
arctan
x
+
C
.
{\displaystyle \int {\frac {{\mathsf {d}}x}{x^{2}+1}}=\arctan x+C.}
∫
d
x
x
2
+
a
2
=
1
a
arctan
x
a
+
C
,
a
≠
0
{\displaystyle \int {\frac {{\mathsf {d}}x}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C,a\not =0}
∫
1
a
2
−
x
2
d
x
=
arcsin
x
a
+
C
,
a
>
0
{\displaystyle \int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\;{\mathsf {d}}x=\arcsin {\frac {x}{a}}+C,\;a>0}
∫
d
x
x
2
−
a
2
=
1
2
a
ln
|
x
−
a
x
+
a
|
+
C
,
a
≠
0
{\displaystyle \int {\frac {{\mathsf {d}}x}{x^{2}-a^{2}}}={\frac {1}{2a}}\ln \left|{\frac {x-a}{x+a}}\right|+C,\;a\not =0}
∫
d
x
x
2
±
a
2
=
ln
|
x
+
x
2
±
a
2
|
+
C
,
a
≠
0
{\displaystyle \int {\frac {{\mathsf {d}}x}{\sqrt {x^{2}\pm a^{2}}}}=\ln \left|x+{\sqrt {x^{2}\pm a^{2}}}\right|+C,\;a\not =0}
∫
a
2
−
x
2
d
x
=
x
2
a
2
−
x
2
+
x
2
a
arcsin
x
a
+
C
{\displaystyle \int {\sqrt {a^{2}-x^{2}}}\;{\mathsf {d}}x={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {x^{2}}{a}}\arcsin {\frac {x}{a}}+C}
∫
x
2
±
a
2
d
x
=
x
2
a
2
±
x
2
±
a
2
2
ln
|
x
+
x
2
±
a
2
|
+
C
{\displaystyle \int {\sqrt {x^{2}\pm a^{2}}}\;{\mathsf {d}}x={\frac {x}{2}}{\sqrt {a^{2}\pm x^{2}}}\pm {\frac {a^{2}}{2}}\ln \left|x+{\sqrt {x^{2}\pm a^{2}}}\right|+C}
∫
a
x
+
b
d
x
=
(
2
b
3
a
+
2
x
3
)
a
x
+
b
+
C
{\displaystyle \int {\sqrt {ax+b}}\;{\mathsf {d}}x=\left({2b \over 3a}+{2x \over 3}\right){\sqrt {ax+b}}+C}
∫
a
x
+
b
d
x
=
2
3
a
(
a
x
+
b
)
3
/
2
+
C
{\displaystyle \int {\sqrt {ax+b}}dx={2 \over 3a}(ax+b)^{3/2}+C}
Trigonometrinių reiškinių integralai
keisti
∫
sin
a
x
d
x
=
−
1
a
cos
a
x
+
C
.
{\displaystyle \int \sin ax\;{\mathsf {d}}x=-{\frac {1}{a}}\cos ax+C.}
∫
cos
a
x
d
x
=
1
a
sin
a
x
+
C
.
{\displaystyle \int \cos ax\;{\mathsf {d}}x={\frac {1}{a}}\sin ax+C.}
∫
tan
x
d
x
=
−
ln
|
cos
x
|
+
C
.
{\displaystyle \int \tan x\;{\mathsf {d}}x=-\ln |\cos x|+C.}
∫
c
t
g
x
d
x
=
ln
|
sin
x
|
+
C
.
{\displaystyle \int ctgx\;{\mathsf {d}}x=\ln |\sin x|+C.}
∫
d
x
sin
x
=
ln
|
tan
x
2
|
+
C
.
{\displaystyle \int {\frac {{\mathsf {d}}x}{\sin x}}=\ln \left|\tan {\frac {x}{2}}\right|+C.}
∫
d
x
cos
x
=
ln
|
tan
(
x
2
+
π
4
)
|
+
C
.
{\displaystyle \int {\frac {{\mathsf {d}}x}{\cos x}}=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|+C.}
∫
d
x
sin
2
x
=
−
cot
x
+
C
=
−
1
tan
x
+
C
=
−
cos
x
sin
x
+
C
.
{\displaystyle \int {\frac {{\mathsf {d}}x}{\sin ^{2}x}}=-\cot x+C=-{\frac {1}{\tan x}}+C=-{\frac {\cos x}{\sin x}}+C.}
∫
d
x
cos
2
x
=
tan
x
+
C
.
{\displaystyle \int {\frac {{\mathsf {d}}x}{\cos ^{2}x}}=\tan x+C.}
∫
sin
n
(
a
x
)
cos
m
(
a
x
)
d
x
=
sin
n
+
1
(
a
x
)
a
(
m
−
1
)
cos
m
−
1
(
a
x
)
−
m
−
m
+
2
m
−
1
∫
sin
n
(
a
x
)
cos
m
−
2
(
a
x
)
d
x
(
m
≠
1
)
,
{\displaystyle \int {\frac {\sin ^{n}(ax)}{\cos ^{m}(ax)}}{\mathsf {d}}x={\frac {\sin ^{n+1}(ax)}{a(m-1)\cos ^{m-1}(ax)}}-{\frac {m-m+2}{m-1}}\int {\frac {\sin ^{n}(ax)}{\cos ^{m-2}(ax)}}{\mathsf {d}}x\quad (m\neq 1),}
∫
sin
n
(
a
x
)
cos
m
(
a
x
)
d
x
=
−
sin
n
−
1
(
a
x
)
a
(
n
−
m
)
cos
m
−
1
(
a
x
)
+
n
−
1
n
−
m
∫
sin
n
−
2
(
a
x
)
cos
m
(
a
x
)
d
x
(
m
≠
n
)
,
{\displaystyle \int {\frac {\sin ^{n}(ax)}{\cos ^{m}(ax)}}{\mathsf {d}}x=-{\frac {\sin ^{n-1}(ax)}{a(n-m)\cos ^{m-1}(ax)}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}(ax)}{\cos ^{m}(ax)}}{\mathsf {d}}x\quad (m\neq n),}
∫
sin
n
(
a
x
)
cos
m
(
a
x
)
d
x
=
sin
n
−
1
(
a
x
)
a
(
m
−
1
)
cos
m
−
1
(
a
x
)
−
n
−
1
m
−
1
∫
sin
n
−
1
(
a
x
)
cos
m
−
2
(
a
x
)
d
x
(
m
≠
1
)
.
{\displaystyle \int {\frac {\sin ^{n}(ax)}{\cos ^{m}(ax)}}{\mathsf {d}}x={\frac {\sin ^{n-1}(ax)}{a(m-1)\cos ^{m-1}(ax)}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-1}(ax)}{\cos ^{m-2}(ax)}}{\mathsf {d}}x\quad (m\neq 1).}
∫
sin
2
(
a
x
)
cos
n
(
a
x
)
d
x
=
sin
(
a
x
)
a
(
n
−
1
)
cos
n
−
1
(
a
x
)
−
1
n
−
1
∫
d
x
cos
n
−
2
(
a
x
)
(
n
≠
1
)
.
{\displaystyle \int {\frac {\sin ^{2}(ax)}{\cos ^{n}(ax)}}{\mathsf {d}}x={\frac {\sin(ax)}{a(n-1)\cos ^{n-1}(ax)}}-{\frac {1}{n-1}}\int {\frac {{\mathsf {d}}x}{\cos ^{n-2}(ax)}}\quad (n\neq 1).}
∫
sin
3
(
a
x
)
cos
n
(
a
x
)
d
x
=
1
a
[
1
(
n
−
1
)
cos
n
−
1
(
a
x
)
−
1
(
n
−
3
)
cos
n
−
3
(
a
x
)
]
(
n
≠
1
,
n
≠
3
)
.
{\displaystyle \int {\frac {\sin ^{3}(ax)}{\cos ^{n}(ax)}}{\mathsf {d}}x={\frac {1}{a}}\left[{\frac {1}{(n-1)\cos ^{n-1}(ax)}}-{\frac {1}{(n-3)\cos ^{n-3}(ax)}}\right]\quad (n\neq 1,\;\;n\neq 3).}
Taip pat skaitykite
keisti
Integravimo metodai
Nuorodos
keisti
integralų lentelė
integralų lentelė su įrodymais
http://www.mathwords.com/i/integral_table.htm
![{\displaystyle \int {\frac {\sin ^{3}(ax)}{\cos ^{n}(ax)}}{\mathsf {d}}x={\frac {1}{a}}\left[{\frac {1}{(n-1)\cos ^{n-1}(ax)}}-{\frac {1}{(n-3)\cos ^{n-3}(ax)}}\right]\quad (n\neq 1,\;\;n\neq 3).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8316dba4cdbb7f91a4c4d9ed385ff8aaac248ef6)
