Mathematics

# Integrate :$\displaystyle\int {\dfrac{{dx}}{{x\left( {{x^3} + 1} \right)}}}$

##### SOLUTION

Consider the given integral.

$I=\displaystyle\int{\dfrac{{{x}^{3}}dx}{{{x}^{4}}\left( {{x}^{3}}+1 \right)}}$

$I=\displaystyle\int{\dfrac{dx}{{{x}^{4}}\left( 1+\dfrac{1}{{{x}^{3}}} \right)}}$

Let $t=1+\dfrac{1}{{{x}^{3}}}$

$\dfrac{dt}{dx}=0-\dfrac{3}{{{x}^{4}}}$

$-\dfrac{dt}{3}=\dfrac{dx}{{{x}^{4}}}$

Therefore,

$I=-\dfrac{1}{3}\displaystyle\int{\dfrac{dt}{t}}$

$I=-\dfrac{1}{3}\ln \left( t \right)+C$

On putting the value of $t$, we get

$I=-\dfrac{1}{3}\ln \left( 1+\dfrac{1}{{{x}^{3}}} \right)+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

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1 Verified Answer | Published on 17th 09, 2020

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