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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