Mathematics

# Evaluate $\overset { 3 }{ \underset { 0 }{\int } } \dfrac{x}{\sqrt{x^2+16}}dx$.

##### SOLUTION
$\int_{0}^{3}{\cfrac{x}{\sqrt{{x}^{2} + 16}} dx}$
Let
${x}^{2} + 16 = t \begin{cases} \text{when } x = 0 \Rightarrow t = 16 \\ \text{when } x = 3 \Rightarrow t = 25 \end{cases}$
Differentiating above equation w.r.t. $x$, we have
$2x \; dx = dt$
$x \; dx = \cfrac{dt}{2}$
Now, the integration will be in the form-
$\int_{16}^{25}{\cfrac{dt}{2 \sqrt{t}}}$
$= \cfrac{1}{2} \int{{t}^{- \frac{1}{2}} \; dt}$
$= \cfrac{1}{2} \left[ \sqrt{t} \right]_{16}^{25}$
$= \cfrac{1}{2} \left[ \sqrt{25} - \sqrt{16} \right]$
$= \cfrac{1}{2} \left( 5 - 4 \right) = \cfrac{1}{2}$
Hence the value of $\int_{0}^{3}{\cfrac{x}{\sqrt{{x}^{2} + 16}} dx}$ is $\cfrac{1}{2}$.

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Subjective Medium Published on 17th 09, 2020
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