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Таблица на основни интеграли
Јазик
Набљудувај
Уреди
Основни интеграли
уреди
∫
x
p
d
x
=
x
p
+
1
p
+
1
+
C
,
p
≠
−
1
{\displaystyle \int x^{p}\,dx={\frac {x^{p+1}}{p+1}}+C,\,\,\,\ p\neq -1}
∫
d
x
x
=
ln
|
x
|
+
C
{\displaystyle \int {\frac {dx}{x}}=\operatorname {ln} |x|+C}
∫
a
x
d
x
=
a
x
ln
x
+
C
,
a
>
0
,
a
≠
1
{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\operatorname {ln} x}}+C,\,\,\,\ a>0,\,\ a\neq 1}
∫
e
x
d
x
=
e
x
+
C
{\displaystyle \int e^{x}\,dx=e^{x}+C}
∫
sin
x
d
x
=
−
cos
x
+
C
{\displaystyle \int \operatorname {sin} x\,dx=-\operatorname {cos} x+C}
∫
cos
x
d
x
=
sin
x
+
C
{\displaystyle \int \operatorname {cos} x\,dx=\operatorname {sin} x+C}
∫
d
x
cos
2
x
=
tg
x
+
C
{\displaystyle \int {\frac {dx}{\operatorname {cos} ^{2}x}}=\operatorname {tg} x+C}
∫
d
x
sin
2
x
=
−
ctg
x
+
C
{\displaystyle \int {\frac {dx}{\operatorname {sin} ^{2}x}}=-\operatorname {ctg} x+C}
∫
d
x
1
−
x
2
=
arcsin
x
+
C
{\displaystyle \int {\frac {dx}{\sqrt {1-x^{2}}}}=\operatorname {arcsin} x+C}
∫
d
x
1
+
x
2
=
arctg
x
+
C
{\displaystyle \int {\frac {dx}{1+x^{2}}}=\operatorname {arctg} x+C}
Останати интеграли
уреди
∫
d
x
a
x
+
b
=
ln
|
a
x
+
b
|
+
C
,
a
≠
0
{\displaystyle \int {\frac {dx}{ax+b}}=\operatorname {ln} |ax+b|+C,\,\,\,\ a\neq 0}
∫
d
x
x
2
+
a
2
=
1
a
arctg
x
a
+
C
{\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\operatorname {arctg} {\frac {x}{a}}+C}
∫
d
x
x
2
−
a
2
=
1
2
a
ln
|
x
−
a
x
+
a
|
+
C
{\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}={\frac {1}{2a}}\operatorname {ln} \left|{\frac {x-a}{x+a}}\right|+C}
∫
d
x
a
x
2
+
b
x
+
c
=
{
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
,
b
2
−
4
a
c
>
0
1
4
a
c
−
b
2
arctg
2
a
x
+
b
4
a
c
−
b
2
+
C
,
b
2
−
4
a
c
<
0
{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\begin{cases}{\frac {1}{\sqrt {b^{2}-4ac}}}\operatorname {ln} \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|+C,&b^{2}-4ac>0\\\\{\frac {1}{\sqrt {4ac-b^{2}}}}\operatorname {arctg} {\frac {2ax+b}{4ac-b^{2}}}+C,&b^{2}-4ac<0\end{cases}}}
∫
d
x
x
2
±
a
2
=
ln
|
x
+
x
2
±
a
2
|
+
C
{\displaystyle \int {\frac {dx}{\sqrt {x^{2}\pm a^{2}}}}=\operatorname {ln} \left|x+{\sqrt {x^{2}\pm a^{2}}}\right|+C}
∫
d
x
a
2
−
x
2
=
sgn
a
⋅
arcsin
x
a
+
C
{\displaystyle \int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\operatorname {sgn} a\cdot \operatorname {arcsin} {\frac {x}{a}}+C}
∫
d
x
a
x
2
+
b
x
+
c
=
1
a
ln
|
x
+
b
2
a
+
x
2
+
b
a
x
+
c
a
|
+
C
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\operatorname {ln} \left|x+{\frac {b}{2a}}+{\sqrt {x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}}\right|+C}
∫
x
2
±
a
2
d
x
=
1
2
[
x
x
2
±
a
2
+
a
2
ln
|
x
+
x
2
±
a
2
|
]
+
C
{\displaystyle \int {\sqrt {x^{2}\pm a^{2}}}\,dx={\frac {1}{2}}\left[x{\sqrt {x^{2}\pm a^{2}}}+a^{2}\operatorname {ln} \left|x+{\sqrt {x^{2}\pm a^{2}}}\right|\right]+C}
∫
a
2
−
x
2
d
x
=
1
2
[
sgn
a
⋅
a
2
arcsin
x
a
+
x
a
2
−
x
2
]
+
C
{\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{2}}\left[\operatorname {sgn} a\cdot a^{2}\operatorname {arcsin} {\frac {x}{a}}+x{\sqrt {a^{2}-x^{2}}}\right]+C}
∫
a
x
2
+
b
x
+
c
d
x
=
a
2
[
(
x
+
b
2
a
)
x
2
+
b
a
x
+
c
a
+
4
a
c
−
b
2
4
a
2
ln
|
x
+
b
2
a
+
x
2
+
b
a
x
+
c
a
|
]
+
C
{\displaystyle \int {\sqrt {ax^{2}+bx+c}}\,dx={\frac {\sqrt {a}}{2}}\left[\left(x+{\frac {b}{2a}}\right){\sqrt {x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}}+{\frac {4ac-b^{2}}{4a^{2}}}\operatorname {ln} \left|x+{\frac {b}{2a}}+{\sqrt {x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}}\right|\right]+C}
![{\displaystyle \int {\sqrt {x^{2}\pm a^{2}}}\,dx={\frac {1}{2}}\left[x{\sqrt {x^{2}\pm a^{2}}}+a^{2}\operatorname {ln} \left|x+{\sqrt {x^{2}\pm a^{2}}}\right|\right]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f41da46a21697d5c7f42211bddb4a6069a2494)
![{\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{2}}\left[\operatorname {sgn} a\cdot a^{2}\operatorname {arcsin} {\frac {x}{a}}+x{\sqrt {a^{2}-x^{2}}}\right]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d7d5f53fa0b0532929298f23db229a1021b17e)
![{\displaystyle \int {\sqrt {ax^{2}+bx+c}}\,dx={\frac {\sqrt {a}}{2}}\left[\left(x+{\frac {b}{2a}}\right){\sqrt {x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}}+{\frac {4ac-b^{2}}{4a^{2}}}\operatorname {ln} \left|x+{\frac {b}{2a}}+{\sqrt {x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}}\right|\right]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f568c5176145c7ca52eb07793557511df3f15450)
