Mathematics

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

##### SOLUTION
Let $I=\displaystyle \int \dfrac{1}{\sqrt x+ \sqrt{x^3}}.dx$
$=\displaystyle \int \dfrac{1}{x^{\dfrac{1}{2}}+x^{\dfrac{3}{2}}} .dx$
Put $x=t^2$
$\therefore dx=2t dt$
Also
$x^{\dfrac{1}{2}}=(t^2)^{\dfrac{1}{2}}=t$
and
$x^{\dfrac{3}{2}}=(t^2)^{\dfrac{3}{2}}=t^3$
$\therefore I=\displaystyle \int \dfrac{2tdt}{t+t^3}$
$=2 \displaystyle \int \dfrac{tdt}{t(1+t^2)}$
$=2 \displaystyle \int \dfrac{1}{(1+t^2)} dt$
$=2 \tan^{-1}t+c$
$=2 \tan^{-1} (\sqrt x)+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

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