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# Find the integral $\int\frac{1}{x\left(x^2+x+1\right)}dx$

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##  Final answer to the problem

$\ln\left|x\right|+\frac{\sqrt{3}\left(-1\right)\arctan\left(\frac{1+2x}{\sqrt{3}}\right)}{3}+\ln\left|\frac{\sqrt{3}}{2\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}}}\right|+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Integrate by partial fractions
• Integrate by substitution
• Integrate by parts
• Integrate using tabular integration
• Integrate by trigonometric substitution
• Weierstrass Substitution
• Integrate using trigonometric identities
• Integrate using basic integrals
• Product of Binomials with Common Term
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1

Rewrite the fraction $\frac{1}{x\left(x^2+x+1\right)}$ in $2$ simpler fractions using partial fraction decomposition

$\frac{1}{x\left(x^2+x+1\right)}=\frac{A}{x}+\frac{Bx+C}{x^2+x+1}$

Learn how to solve integrals by partial fraction expansion problems step by step online.

$\frac{1}{x\left(x^2+x+1\right)}=\frac{A}{x}+\frac{Bx+C}{x^2+x+1}$

Learn how to solve integrals by partial fraction expansion problems step by step online. Find the integral int(1/(x(x^2+x+1)))dx. Rewrite the fraction \frac{1}{x\left(x^2+x+1\right)} in 2 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C. The first step is to multiply both sides of the equation from the previous step by x\left(x^2+x+1\right). Multiplying polynomials. Simplifying.

##  Final answer to the problem

$\ln\left|x\right|+\frac{\sqrt{3}\left(-1\right)\arctan\left(\frac{1+2x}{\sqrt{3}}\right)}{3}+\ln\left|\frac{\sqrt{3}}{2\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}}}\right|+C_0$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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

e
π
ln
log
log
lim
d/dx
Dx
|◻|
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

###  Main Topic: Integrals by Partial Fraction Expansion

The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.