Integrate the function: 1 x+a+x+b d x - Vidyadip

Question

Integrate the function: $\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} d x$

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Answer

Let I = $\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} d x$. Then,
$I=\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} \times \frac{\sqrt{x+a}-\sqrt{x+b}}{\sqrt{x+a}-\sqrt{x+b}} \times d x$
$=\int \frac{\sqrt{x+a}-\sqrt{x+b}}{x+a-x-b} \times d x$
$=\int \frac{\sqrt{x+a}-\sqrt{x+b}}{a-b} \times d x$
$=\frac{1}{a-b}\left[\frac{2}{3}(x+a)^{\frac{3}{2}}-\frac{2}{3}(x+b)^{\frac{3}{2}}\right]+c$
$=\frac{2}{3(a-b)}\left[(x+a)^{\frac{3}{2}}-(x+b)^{\frac{3}{2}}\right]+c$
$\therefore \ I=\frac{2}{3(a-b)}\left[(x+a)^{\frac{3}{2}}-(x+b)^{\frac{3}{2}}\right]+c$ .

Which is the required solution.

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