Mathematics

# Integrate with respect to $x$:$\dfrac {x}{\sqrt {x+2}}$

##### SOLUTION

Consider the given integral.

$I=\int{\dfrac{x}{\sqrt{x+2}}dx}$

Put,

$u=x+2\Rightarrow du=dx$

Therefore,

$I=\int{\dfrac{u-2}{\sqrt{u}}du}$

$I=\int{\left( \sqrt{u}-\dfrac{2}{\sqrt{u}} \right)du}$

$I=\int{\sqrt{u}du}-2\int{\dfrac{1}{\sqrt{u}}du}$

$I=\dfrac{2}{3}{{\left( u \right)}^{3/2}}-4\sqrt{u}+C$

Now, return the value of $x$.

$I=\dfrac{2}{3}{{\left( x+2 \right)}^{3/2}}-4\sqrt{x+2}+C$

$I=\dfrac{2\sqrt{x+2}\left( x+2-6 \right)}{3}+C$

$I=\dfrac{2\sqrt{x+2}\left( x-4 \right)}{3}+C$

Hence, this is the required value of the integral.

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

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