Question

Solve:
$\int \frac{\sqrt{x}-\sqrt{a}}{\sqrt{x+a}}dx$

Answer 1

Given,

$\int \frac{\sqrt{x}-\sqrt{a}}{\sqrt{x+a}}dx$

apply integration by parts,

$u=\sqrt{x}-\sqrt{a}$

$v^{\prime }=\frac{1}{\sqrt{x+a}}$

$=2(\sqrt{x}-\sqrt{a})\sqrt{x+a}-\int \sqrt{\frac{x+a}{x}}dx$

$2(\sqrt{x}-\sqrt{a})\sqrt{x+a}-(\sqrt{x}\sqrt{a+x}+a\ln |\sqrt{\frac{x}{a}}+\sqrt{\frac{a+x}{a}}|)$

$=\sqrt{x}\sqrt{x+a}-a\ln |\sqrt{\frac{x}{a}}+\sqrt{\frac{x+a}{a}}|-2\sqrt{a}\sqrt{x+a}$

$=\sqrt{x}\sqrt{x+a}-a\ln |\sqrt{\frac{x}{a}}+\sqrt{\frac{x+a}{a}}|-2\sqrt{a}\sqrt{x+a}+c$