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Find the integral $\int\frac{x}{3x^2-1}dx$

Used Formulas

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Solution

Formulas

Videos

Basic Derivatives

· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$

Integration Techniques

· Integration by Substitution
$\int f\left(x\right)dx=\int f\left(g\left(t\right)\right) g'\left(t\right)dt$
· Integration by Parts
$\int udv=uv - \int vdu$

Special Products

· Difference of two Squares
$a^2-b^2=(a+b)(a-b)$

Basic Integrals

· Sum Rule for Integration
$\int\left(a+b+...\right)dx=\int adx+\int bdx+...$
· Constant factor Rule
$\int cxdx=c\int xdx$

Integrals of Rational Functions

$\int\frac{n}{x+b}dx=nsign\left(x\right)\ln\left(x+b\right)+C$

Function Plot

Plotting: $-\frac{\sqrt{3}}{6}x\ln\left(\sqrt{3}x+1\right)+\frac{\sqrt{3}}{6}x\ln\left(\sqrt{3}x-1\right)+\frac{\sqrt{3}}{6}\left(\left(\sqrt{3}x+1\right)\ln\left(\sqrt{3}x+1\right)-\sqrt{3}x-1\right)-\frac{\sqrt{3}}{6}\left(\left(\sqrt{3}x-1\right)\ln\left(\sqrt{3}x-1\right)-\sqrt{3}x+1\right)+C_0$

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7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

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Main Topic: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

Used Formulas

See formulas (7)

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