Mathematics

# Evaluate:$\displaystyle \int \dfrac {dx}{x(x^{3}+8)}$

##### SOLUTION
We have,
$I=\int { \dfrac { dx }{ x\left( { x }^{ 3 }+8 \right) } }$
$I=\int { \dfrac { { x }^{ 2 }dx }{ { x }^{ 3 }\left( { x }^{ 3 }+8 \right) } }$

Let ${ x }^{ 3 }=t\quad \Rightarrow 3{ x }^{ 2 }dx=dt$
$\Rightarrow { x }^{ 2 }dx=\dfrac { dt }{ 3 }$

$\therefore$    $I=\int { \dfrac { 1 }{ 3t\left( t+8 \right) } } dt$
$=\dfrac { 1 }{ 3 } \int { \dfrac { 1 }{ 8 } \dfrac { \left( t+8-t \right) }{ \left( t \right) \left( t+8 \right) } } dt$
$=\dfrac { 1 }{ 24 } \int { \left( \dfrac { 1 }{ t } -\dfrac { 1 }{ t+8 } \right) } dt$
$=\dfrac { 1 }{ 24 } \log\left( t \right) -log\left( t+8 \right) +c$
$=\dfrac { 1 }{ 24 } \log\left| \dfrac { t }{ t+8 } \right| +c$
$=\dfrac { 1 }{ 24 } \log\left| \dfrac { { x }^{ 3 } }{ { x }^{ 3 }+8 } \right| +c$

Hence, this is the answer.

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