Mathematics

Evaluate :
$$\int { [\sqrt { tanx } +\sqrt { cotx } ] } dx$$.


SOLUTION
$$I= \displaystyle\int (\sqrt{\tan x}+\sqrt{\cot x})dx$$

$$\therefore I= \displaystyle\int \sqrt{\dfrac{\sin x}{\cos x}}+\sqrt{\dfrac{\cos x}{\sin x}}dx$$

$$\therefore I= \displaystyle\int \dfrac{\sin x+ \cos x}{\sqrt{\sin x \cos x}}dx$$
$$ \sin x- \cos x =t$$

$$\therefore (\cos x+ \sin x)dx=dt$$

$$\therefore 1-2 \sin x \cos x =t^{2}$$

$$\therefore I= \displaystyle\int \dfrac{\sqrt{2}}{\sqrt{1-t^{2}}}dt$$

$$\therefore I=\sqrt{2}\sin^{-1}t+c$$

$$\therefore I=\sqrt{2} \sin^{-1}(\sin x- \cos x)+c$$
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Subjective Medium Published on 17th 09, 2020
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