Question

Integrate the following functions w.r.t. x.

$\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}}dx.$

Answer 1

$\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}}dx=\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}}\times \frac{\sqrt{x+a}-\sqrt{x+b}}{\sqrt{x+a}-\sqrt{x+b}}dx=\int \frac{\sqrt{x+a}-\sqrt{x+b}}{\sqrt{x+a}-\sqrt{x+b}}dx=\frac{1}{a-b}\int \left\{\right(x+a{)}^{1/2}-(x+b{)}^{1/2}\}dx=\frac{1}{a-b}\{\frac{(x+a{)}^{\left(1/2\right)+1}}{\frac{1}{2}+1}-\frac{(x+b{)}^{\left(1/2\right)+1}}{\frac{1}{2}+1}\}+C=\frac{1}{a-b}\{\frac{(x+a{)}^{3/2}}{3/2}-\frac{(x+b{)}^{3/2}}{3/2}\}+C=\frac{2}{3(a-b)}\left[\right(x+a{)}^{3/2}-(x+b{)}^{3/2}]+C$