Find $\frac{\mathrm{d}}{\mathrm{d}x}\left( {{e}^{{{\sin }^{-1}}\left( {{x}^{2}} \right)}} \right)$

Question

Find $\frac{\mathrm{d}}{\mathrm{d}x}\left( {{e}^{{{\sin }^{-1}}\left( {{x}^{2}} \right)}} \right)$ and hence find the exact value of $\displaystyle\int_{0}^{1}{\frac{x{{\cos }^{-1}}\left( {{x}^{2}} \right)}{\sqrt{1-{{x}^{4}}}}}\left( {{e}^{{{\sin }^{-1}}\left( {{x}^{2}} \right)}} \right)\mathrm{d}x$.

Accepted Answer

Derivative $\frac{2xe^{\sin^{-1}(x^2)}}{\sqrt{1-x^4}}$, $-1<x<1$; integral $\frac12\left(e^{\frac\pi2}-1\right)-\frac\pi4$.

2014 TPJC P1 Q4

2014 TPJC P1 Q4

Solution:

Solution locked