Mathematics

# Evaluate the following integrals:$\displaystyle \int { \cfrac { x }{ \left( { x }^{ 2 }+4 \right) \sqrt { { x }^{ 2 }+9 } } } dx\quad$

##### SOLUTION
Now,
$\displaystyle\int { \cfrac { x }{ \left( { x }^{ 2 }+4 \right) \sqrt { { x }^{ 2 }+9 } } } dx\quad$

Put $x^2+9=p^2$......(1).

or, $2x\ dx=2p\ dp$.

Now using these in the above expression we get,

$=\displaystyle\int { \cfrac { p }{\left( { p }^{ 2 }+5 \right) p } } dp\quad$

$=\displaystyle\int { \cfrac { 1 }{ \left( { p }^{ 2 }+5 \right) } } dp\quad$

$=\displaystyle\int { \cfrac { 1 }{ \left( { p }^{ 2 }+(\sqrt{5})^2 \right) } } dp\quad$

$=\dfrac{1}{2\sqrt{5}}\log\left|\dfrac{p-\sqrt{5}}{p+\sqrt{5}}\right|+c$

$=\dfrac{1}{2\sqrt{5}}\log\left|\dfrac{\sqrt{x^2+9}-\sqrt{5}}{\sqrt{x^2+9}+\sqrt{5}}\right|+c$ [ Using (1)]

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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