Mathematics

$$\int \dfrac {\sin^{-1}x-\cos^{-1}x}{\sin^{-1}x+\cos^{-1}x}dx=$$


ANSWER

$$\dfrac {2}{\pi}[x \sin^{-1}x-x \cos^{-1}x+2\sqrt {1-x^{2}}]+c$$


SOLUTION
$$\displaystyle \int{\dfrac{\sin^{-1}x-\cos^{-1}x}{\sin^{-1}x+\cos^{-1}x}}dx$$
$$\sin^{-1}x+\cos^{-1}x=\pi /2$$
$$\dfrac{2}{\pi}\displaystyle \int{\sin^{-1}x}\displaystyle \int{\cos^{-1}x}dx$$
$$\Rightarrow \dfrac{2}{\pi}\left[x\sin^{-1}x+\sqrt{1-x^2}-x\cos^{-1}x+\sqrt{1-x^2}\right]+c$$
$$=\dfrac{2}{\pi}[x\sin^{-1}x-x\cos^{-1}x+2\sqrt{1-x^2}]+c$$
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Single Correct Medium Published on 17th 09, 2020
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