Evaluate:
Question:

Evaluate: $\int \frac{\mathrm{x}}{\sqrt{\mathrm{x}+\mathrm{a}}-\sqrt{\mathrm{x}+\mathrm{b}}} \mathrm{dx}$

Solution:

Rationalise the given equation we get

$\Rightarrow \int \frac{x}{\sqrt{x+a}-\sqrt{x-b}} \times \frac{\sqrt{x+a}+\sqrt{x-b}}{\sqrt{x+a}+\sqrt{x-b}} d x$

$\Rightarrow \int \frac{x(\sqrt{x+a}-\sqrt{x-b})}{x+a-x-b} d x$

$\Rightarrow \int \frac{x(\sqrt{x+a}-\sqrt{x-b})}{a-b} d x$

$\Rightarrow \frac{1}{a-b} \int x(\sqrt{x+a}-\sqrt{x-b}) d x$

Assume $x=\sqrt{t}$

$\Rightarrow \mathrm{dx}=\frac{\mathrm{dt}}{2 \sqrt{t}}$

Substituting $t$ and $d t$

$\Rightarrow \int \sqrt{\mathrm{t}} \frac{(\sqrt{\sqrt{t}+a}-\sqrt{\sqrt{t}-b})}{2 \sqrt{t}(a-b)} d t$

$\Rightarrow \frac{1}{2(a-b)} \int(\sqrt{\sqrt{t}+a}-\sqrt{\sqrt{t}-b}) d t$

$\Rightarrow \frac{1}{2(a-b)} \int(\sqrt{t}+a)^{1 \backslash 2} d t-\int(\sqrt{t}-b)^{1 / 2} d t$

$\Rightarrow \frac{1}{2(a-b)}\left(\frac{4}{3}\left(\sqrt{t}+a^{2}\right)^{\frac{2}{2}}-\frac{4}{3}\left(t-a^{2}\right)^{\frac{3}{2}}\right)$

But $x=\sqrt{t}$

$\Rightarrow \frac{1}{2(a-b)}\left(\frac{2}{3}(x+a)^{\frac{3}{2}}-\frac{2}{3}(x-b)^{\frac{3}{2}}\right)$