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
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