Question

If $\Phi \left(x\right)={lim}_{n\to \infty }\frac{x^n-x^{-n}}{x^n+x^{-n}},0<x<1,ϵN$, then $\int \left(si{n}^{-1}x\right)\left(\Phi \right(x\left)\right)dx$ is equal to

• A. $xsi{n}^{-1}\left(x\right)+\sqrt{1-x^2}+C$
• B. $-\left(xsi{n}^{-1}\right(x)+\sqrt{1-x^2})$
• C. $xsi{n}^{-1x+\sqrt{1-x^2}+C}$
• D. None of these

Answer 1

The correct option is A $xsi{n}^{-1}\left(x\right)+\sqrt{1-x^2}+C$

We have $\varphi \left(x\right)={lim}_{n\to \infty }\frac{x^{2n}-1}{x^{2n}+1}=1,0<x<1\therefore \int si{n}^{-1}x.\varphi \left(x\right)dx=\int si{n}^{-1}xdx=[xsi{n}^{-1}x+\sqrt{1-x^2}]+c$