Integrate the following functions w.r.t. x: 1 x+x^3 - Vidyadip

Question

Integrate the following functions w.r.t. x:
$\frac{1}{\sqrt{x}+\sqrt{x^3}}$

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Answer

Let $I=\int \frac{1}{\sqrt{x}+\sqrt{x^3}} d x$$=\int \frac{1}{x^{\frac{1}{2}}+x^{\frac{3}{2}}} d x$
Put $x=t^2 \quad \therefore d x=2 t d t$
Also $x^{\frac{1}{2}}=\left(t^2\right)^{\frac{1}{2}}=t$ and $x^{\frac{3}{2}}=\left(t^2\right)^{\frac{3}{2}}=t^3$
$\therefore I =\int \frac{2 t d t}{t+t^3}$
$ =2 \int \frac{t d t}{t\left(1+t^2\right)}$
$=2 \int \frac{1}{1+t^2} d t$
$=2 \tan ^{-1} t+c$
$=2 \tan ^{-1}(\sqrt{x})+c .$

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