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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

$\ln\left|x\right|+\frac{-\sqrt{3}\arctan\left(\frac{1+2x}{\sqrt{3}}\right)}{3}+\ln\left|\frac{\sqrt{3}}{2\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}}}\right|+C_0$
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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
Can't find a method? Tell us so we can add it.
1

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

$\frac{1}{x}+\frac{-x-1}{x^2+x+1}$

Learn how to solve problems step by step online.

$\frac{1}{x}+\frac{-x-1}{x^2+x+1}$

Learn how to solve 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. Expand the integral \int\left(\frac{1}{x}+\frac{-x-1}{x^2+x+1}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{x}dx results in: \ln\left|x\right|. The integral \int\frac{-x-1}{x^2+x+1}dx results in: -\int\frac{x+1}{x^2+x+1}dx.

##  Final answer to the problem

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

##  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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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
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