Question

find $\displaystyle\sf \int \dfrac{sin^{-1}\sqrt{x}-cos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}} dx \;\; ; \;\; x \in [0,1]$

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Answer 1

$\sf{\underbrace{\underline{ understanding\:the\: Question}}}$

firstly use the identity $\sf sin^{-1} \theta +cos^{ -1} \theta = \dfrac{\pi}{2}$ to convert integrand in terms of $\sf sin^{-1}$ only. then, integrate by using substitution method.

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Answer:-

let $\sf I = \displaystyle\sf \int \dfrac{sin^{-1}\sqrt{x}-cos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}} dx$

we know that $\sf sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}=\dfrac{\pi}{2}$

$\sf \implies cos^{-1}\sqrt{x} = \dfrac{\pi}{2} - sin^{-1}\sqrt{x}$

$\displaystyle\sf \therefore I = \int \dfrac{sin^{-1}\sqrt{x}- \left(\dfrac{\pi}{2} - sin^{-1}\sqrt{x} \right)}{\dfrac{\pi}{2}} dx$

$\displaystyle\sf \therefore \int \dfrac{2\:sin^{-1}\sqrt{x}-\dfrac{\pi}{2}}{\dfrac{\pi}{2}} dx = \dfrac{2}{\pi} \int \left( 2\:sin^{-1}\sqrt{x}-\dfrac{\pi}{2}\right) dx$

$\displaystyle\sf = \dfrac{4}{\pi} \int sin^{-1}\sqrt{x} dx - \int 1 \: dx$

$\displaystyle\sf = \dfrac{4}{\pi} \int sin^{-1}\sqrt{x} dx - x$

$\sf \implies I = \dfrac{4}{\pi} I_1 - x + C\:\:\:\:\;\:\dots(1)$

where $\displaystyle\sf I_1 = \int sin^{-1}\sqrt{x}dx$

Put $\sf \sqrt{x} = tz$

• $\sf x = t^2$
• $\sf dx = 2t\:dt$

$\displaystyle\sf I_1 = \int sin^{-1}t+2t\:dt$

$\displaystyle\sf = 2 \int sin^{-1} t \cdot t \: dt$

$\displaystyle\sf = 2 \left[ sin^{-1}\cdot \dfrac{t^2}{2} - \int \dfrac{1}{\sqrt{1-t^2}} \cdot \dfrac{t^2}{2} dt\right]$

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

$\displaystyle\sf = t^2\:sin^{-1}\:t - \int \dfrac{(1-t^2)+1}{\sqrt{1-t^2}} dt$

$\displaystyle\sf = t^2\:sin^{-1}t+t\sqrt{1-t^2} + \dfrac{1}{2} sin^{-1}\:t - sin^{-1}\:t \left( t^2 - \dfrac{1}{t} \right) \: sin^{-1}\:t + \dfrac{1}{2} \: r \sqrt{1-t^2}$

$\sf = \dfrac{1}{2} \left[ (2x-1)\:sin^{-1}\:\sqrt{x} + \sqrt{x}\sqrt{1-x}\right]$

$\sf = \dfrac{1}{2} \left[ (2x-1)\:sin^{-1}\:\sqrt{x} + \sqrt{x-x^2}\right]$

on putting values of $\sf I_1$ in eq. (i) , we get value of $\displaystyle\sf \int \dfrac{sin^{-1}\sqrt{x}-cos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}} dx$ as

$\boxed{\sf = \dfrac{2}{\pi} \left[ (2x-1)sin^{-1}\:\sqrt{x} + \sqrt{x-x^2}\right] - x + C}$