Mathematics

Integrate the following function with respect to $$x$$
$$\dfrac{x.\sec^{2}{x^{2}}}{\sqrt{\tan^{3}{(x^{2})}}}$$


SOLUTION
Let $$ \displaystyle I= \int \dfrac{x.\sec^{2}{x^{2}}}{\sqrt{\tan^{3}{(x^{2})}}}.dx$$
Put $$\tan (x^2)=t$$
$$\therefore \sec^2 (x^2) \times 2x dx=dt$$
$$\therefore x.sec^2(x^2)dx=\dfrac{dt}{2}$$
$$\therefore I= \displaystyle \int \dfrac{1}{\sqrt{t^3}}.\dfrac{dt}{2}$$
$$=\displaystyle  \dfrac{1}{2} \int t^{-\dfrac{3}{2}}dt$$
$$=\dfrac{1}{2} $$$$\dfrac{t^{-\dfrac{1}{2}}}{-\dfrac{1}{2}}+c$$
$$\dfrac{-1}{\sqrt t} +c$$
$$\dfrac{-1}{\sqrt{\tan(x^2)}} +c$$
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Subjective Medium Published on 17th 09, 2020
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