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Integrals of Rational Functions
Integrals of Rational Functions
We found
28
formulas for integrals of rational functions
1.
∫
a
d
x
=
a
x
+
C
2
\int a dx=ax+C2
∫
a
d
x
=
a
x
+
C
2
Indefinite Integral
Rational Function
2.
∫
x
2
d
x
=
x
2
2
+
C
\int x^2 dx=\dfrac{x^2}{2}+C
∫
x
2
d
x
=
2
x
2
+
C
Indefinite Integral
Rational Function
3.
∫
x
2
d
x
=
x
3
3
+
C
\int x^2 dx=\dfrac{x^3}{3}+C
∫
x
2
d
x
=
3
x
3
+
C
Indefinite Integral
Rational Function
4.
∫
x
p
d
x
=
x
p
+
1
p
+
1
+
C
,
p
≠
1
\int x^p dx=\dfrac{x^{p+1}}{p+1}+C,p \ne 1
∫
x
p
d
x
=
p
+
1
x
p
+
1
+
C
,
p
=
1
Indefinite Integral
Rational Function
5.
∫
(
a
x
+
b
)
n
d
x
=
(
a
x
+
b
)
n
+
1
a
(
n
+
1
)
+
C
,
n
≠
1
\int \left(ax+b\right)^n \,dx=\dfrac{\left(ax+b\right)^{n+1}}{a(n+1)}+C, n \ne 1
∫
(
a
x
+
b
)
n
d
x
=
a
(
n
+
1
)
(
a
x
+
b
)
n
+
1
+
C
,
n
=
1
Indefinite Integral
Rational Function
6.
∫
d
x
x
=
ln
∣
x
∣
+
C
\int \dfrac{dx}{x} = \ln|x|+C
∫
x
d
x
=
ln
∣
x
∣
+
C
Indefinite Integral
Rational Function
7.
∫
1
a
x
+
b
d
x
=
1
a
ln
∣
a
x
+
b
∣
+
C
\int \dfrac{1}{ax+b}dx=\dfrac{1}{a}\ln|ax+b|+C
∫
a
x
+
b
1
d
x
=
a
1
ln
∣
a
x
+
b
∣
+
C
Indefinite Integral
Rational Function
8.
∫
a
x
+
b
c
x
+
d
d
x
=
a
c
x
+
b
c
−
a
d
c
2
ln
∣
c
x
+
d
∣
+
C
\int \dfrac{ax+b}{cx+d}dx=\dfrac{a}{c}x+\dfrac{bc-ad}{c^2} \ln|cx+d|+C
∫
c
x
+
d
a
x
+
b
d
x
=
c
a
x
+
c
2
b
c
−
a
d
ln
∣
c
x
+
d
∣
+
C
Indefinite Integral
Rational Function
9.
∫
d
x
(
x
+
a
)
(
x
+
b
)
d
x
=
1
a
−
b
+
ln
∣
x
+
b
x
+
a
∣
+
C
,
a
≠
b
\int \dfrac{dx}{(x+a)(x+b)} dx=\dfrac{1}{a-b}+\ln\left| \dfrac{x+b}{x+a}\right|+C, a\ne b
∫
(
x
+
a
)
(
x
+
b
)
d
x
d
x
=
a
−
b
1
+
ln
x
+
a
x
+
b
+
C
,
a
=
b
Indefinite Integral
Rational Function
10.
∫
x
a
+
b
x
d
x
=
1
b
2
(
a
+
b
x
−
a
ln
∣
a
+
b
x
∣
)
+
C
\int \dfrac{x}{a+bx}dx=\dfrac{1}{b^2}\left(a+bx-a\ln|a+bx|\right) + C
∫
a
+
b
x
x
d
x
=
b
2
1
(
a
+
b
x
−
a
ln
∣
a
+
b
x
∣
)
+
C
Indefinite Integral
Rational Function
11.
∫
x
2
d
x
a
+
b
x
=
1
b
3
[
1
2
(
a
+
b
x
)
2
−
2
a
(
a
+
b
x
)
+
a
2
ln
∣
a
+
b
x
∣
]
+
C
\int \dfrac{x^2 dx}{a+bx}=\dfrac{1}{b^3}\left[ \dfrac{1}{2} (a+bx)^2 - 2a(a+bx)+a^2\ln|a+bx|\right]+C
∫
a
+
b
x
x
2
d
x
=
b
3
1
[
2
1
(
a
+
b
x
)
2
−
2
a
(
a
+
b
x
)
+
a
2
ln
∣
a
+
b
x
∣
]
+
C
Indefinite Integral
Rational Function
12.
∫
d
x
x
(
a
+
b
x
)
d
x
=
1
a
ln
∣
a
+
b
x
x
∣
+
C
\int \dfrac{dx}{x(a+bx)}dx=\dfrac{1}{a} \ln \left| \dfrac{a+bx}{x} \right| + C
∫
x
(
a
+
b
x
)
d
x
d
x
=
a
1
ln
x
a
+
b
x
+
C
Indefinite Integral
Rational Function
13.
∫
d
x
x
2
(
a
+
b
x
)
d
x
=
−
1
a
x
+
b
a
2
ln
∣
a
+
b
x
x
∣
+
C
\int \dfrac{dx}{x^2(a+bx)}dx=-\dfrac{1}{ax}+\dfrac{b}{a^2} \ln\left|\dfrac{a+bx}{x}\right|+C
∫
x
2
(
a
+
b
x
)
d
x
d
x
=
−
a
x
1
+
a
2
b
ln
x
a
+
b
x
+
C
Indefinite Integral
Rational Function
14.
∫
x
d
x
(
a
+
b
x
)
2
=
1
b
2
(
ln
∣
a
+
b
x
∣
+
a
a
+
b
x
)
+
C
\int \dfrac{xdx}{\left(a+bx\right)^2}=\dfrac{1}{b^2} \left( \ln \left|a+bx\right|+\dfrac{a}{a+bx}\right) + C
∫
(
a
+
b
x
)
2
x
d
x
=
b
2
1
(
ln
∣
a
+
b
x
∣
+
a
+
b
x
a
)
+
C
Indefinite Integral
Rational Function
15.
∫
x
2
(
a
+
b
x
)
2
=
1
b
3
(
a
+
b
x
−
2
a
ln
∣
a
+
b
x
∣
−
a
2
a
+
b
x
)
+
C
\int \dfrac{x^2}{\left(a+bx\right)^2}=\dfrac{1}{b^3} \left(a+bx-2a\ln|a+bx|-\dfrac{a^2}{a+bx}\right)+C
∫
(
a
+
b
x
)
2
x
2
=
b
3
1
(
a
+
b
x
−
2
a
ln
∣
a
+
b
x
∣
−
a
+
b
x
a
2
)
+
C
Indefinite Integral
Rational Function
16.
∫
d
x
x
(
a
+
b
x
)
2
=
1
a
(
a
+
b
x
)
+
1
a
2
ln
∣
a
+
b
x
x
∣
+
C
\int \dfrac{dx}{x(a+bx)^2}=\dfrac{1}{a(a+bx)}+\dfrac{1}{a^2}\ln\left|\dfrac{a+bx}{x}\right|+C
∫
x
(
a
+
b
x
)
2
d
x
=
a
(
a
+
b
x
)
1
+
a
2
1
ln
x
a
+
b
x
+
C
Indefinite Integral
Rational Function
17.
∫
d
x
x
2
−
1
=
1
2
ln
∣
x
−
1
x
+
1
∣
+
C
\int \dfrac{dx}{x^2-1}=\dfrac{1}{2}\ln\left|\dfrac{x-1}{x+1}\right|+C
∫
x
2
−
1
d
x
=
2
1
ln
x
+
1
x
−
1
+
C
Indefinite Integral
Rational Function
18.
∫
d
x
1
−
x
2
=
1
2
ln
∣
1
+
x
1
−
x
∣
+
C
\int \dfrac{dx}{1-x^2}=\dfrac{1}{2}\ln\left|\dfrac{1+x}{1-x}\right|+C
∫
1
−
x
2
d
x
=
2
1
ln
1
−
x
1
+
x
+
C
Indefinite Integral
Rational Function
19.
∫
d
x
a
2
−
x
2
=
1
2
a
ln
∣
a
+
x
a
−
x
∣
+
C
\int \dfrac{dx}{a^2-x^2}=\dfrac{1}{2a}\ln\left|\dfrac{a+x}{a-x}\right|+C
∫
a
2
−
x
2
d
x
=
2
a
1
ln
a
−
x
a
+
x
+
C
Indefinite Integral
Rational Function
20.
∫
d
x
x
2
−
a
2
=
1
2
a
ln
∣
x
−
a
x
+
a
∣
+
C
\int \dfrac{dx}{x^2-a^2}=\dfrac{1}{2a}\ln\left|\dfrac{x-a}{x+a}\right|+C
∫
x
2
−
a
2
d
x
=
2
a
1
ln
x
+
a
x
−
a
+
C
Indefinite Integral
Rational Function
21.
∫
d
x
1
+
x
2
=
tan
−
1
x
+
C
\int \dfrac{dx}{1+x^2}=\tan^{-1}x+C
∫
1
+
x
2
d
x
=
tan
−
1
x
+
C
Indefinite Integral
Rational Function
22.
∫
d
x
a
2
+
x
2
=
1
a
tan
−
1
x
a
+
C
\int \dfrac{dx}{a^2+x^2}=\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+C
∫
a
2
+
x
2
d
x
=
a
1
tan
−
1
a
x
+
C
Indefinite Integral
Rational Function
23.
∫
d
x
a
+
b
x
2
=
1
a
b
arctan
(
x
a
b
)
+
C
,
a
b
>
0
\int \dfrac{dx}{a+bx^2}=\dfrac{1}{\sqrt{ab}}\arctan \left(x\sqrt{\dfrac{a}{b}}\right)+C, ab>0
∫
a
+
b
x
2
d
x
=
ab
1
arctan
(
x
b
a
)
+
C
,
ab
>
0
Indefinite Integral
Rational Function
24.
∫
x
d
x
a
+
b
x
2
=
1
2
b
ln
∣
x
2
+
a
b
∣
+
C
\int \dfrac{xdx}{a+bx^2}=\dfrac{1}{2b}\ln\left|x^2+\frac{a}{b}\right|+C
∫
a
+
b
x
2
x
d
x
=
2
b
1
ln
x
2
+
b
a
+
C
Indefinite Integral
Rational Function
25.
∫
d
x
x
(
a
+
b
x
2
)
=
1
2
a
ln
∣
x
2
a
+
b
x
2
∣
+
C
\int \dfrac{dx}{x\left(a+bx^2\right)}=\dfrac{1}{2a}\ln\left|\dfrac{x^2}{a+bx^2}\right|+C
∫
x
(
a
+
b
x
2
)
d
x
=
2
a
1
ln
a
+
b
x
2
x
2
+
C
Indefinite Integral
Rational Function
26.
∫
d
x
a
2
−
b
2
x
2
=
1
2
a
b
ln
∣
a
+
b
x
a
−
b
x
∣
+
C
\int \frac{dx}{a^2-b^2x^2}=\dfrac{1}{2ab}\ln\left|\dfrac{a+bx}{a-bx}\right|+C
∫
a
2
−
b
2
x
2
d
x
=
2
ab
1
ln
a
−
b
x
a
+
b
x
+
C
Indefinite Integral
Rational Function
27.
∫
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
\int \dfrac{dx}{ax^2+bx+c}=\dfrac{1}{\sqrt{b^2-4ac}}\ln\left| \dfrac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}} \right|+C, b^2-4ac>0
∫
a
x
2
+
b
x
+
c
d
x
=
b
2
−
4
a
c
1
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
Indefinite Integral
Rational Function
28.
∫
d
x
a
x
2
+
b
x
+
c
=
2
4
a
c
−
b
2
arctan
2
a
x
+
b
4
a
c
−
b
2
+
C
\int \dfrac{dx}{ax^2+bx+c}=\dfrac{2}{\sqrt{4ac-b^2}} \arctan \dfrac{2ax+b}{\sqrt{4ac-b^2}}+C
∫
a
x
2
+
b
x
+
c
d
x
=
4
a
c
−
b
2
2
arctan
4
a
c
−
b
2
2
a
x
+
b
+
C
Indefinite Integral
Rational Function
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Isoscales triangle
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