Evaluate: x x+z x+z d x - Vidyadip

Question

Evaluate: $\int \frac{x}{\sqrt{x+z} \sqrt{x+z}} d x$

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Answer

Rationalize the given integrand we get
$ \Rightarrow \int \frac{x}{\sqrt{x+a}-\sqrt{x-b}} \times \frac{\sqrt{x+a}+\sqrt{x-b}}{\sqrt{x+a}+\sqrt{x-b}} dx$
$\Rightarrow \int \frac{x(\sqrt{x+a}-\sqrt{x-b})}{x+a-x-b} d x$
$\Rightarrow \int \frac{x(\sqrt{x+a}-\sqrt{x-b})}{a-b} d x$
$\Rightarrow \frac{1}{a-b} \int x(\sqrt{x+a}-\sqrt{x-b}) d x$
Assume $x =\sqrt{t}$
$\Rightarrow dx=\frac{dt}{2 \sqrt{t}}$
Substituting values of $t $, and $dt ,$
$\Rightarrow \int \sqrt{t} \frac{(\sqrt{\sqrt{t}+a}-\sqrt{\sqrt{t}-b)}}{2 \sqrt{t}(a-b)} d t$
$\Rightarrow \frac{1}{2(a-b)} \int(\sqrt{\sqrt{t}+a}-\sqrt{\sqrt{t}-b}) d t$
$\Rightarrow \frac{1}{2(a-b)} \int(\sqrt{t}+a)^{1 / 2} d t-\int(\sqrt{t}-b)^{1 / 2} d t$
$\Rightarrow \frac{1}{2(a-b)}\left(\frac{4}{3}\left(\sqrt{t}+a^2\right)^{\frac{3}{2}}-\frac{4}{3}\left(t-a^2\right)^{\frac{3}{2}}\right) $
$\text { now replacing } x =\sqrt{t}$
$\Rightarrow \frac{1}{2(a-b)}\left(\frac{2}{3}(x+a)^{\frac{3}{2}}-\frac{2}{3}(x-b)^{\frac{3}{2}}\right)$

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