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 $\displaystyle\int\nolimits_{0}^{1}{\frac{x{{\cos }^{-1}}\left( {{x}^{2}} \right)}{\sqrt{1-{{x}^{4}}}}}\left( {{e}^{{{\sin }^{-1}}\left( {{x}^{2}} \right)}} \right)\mathrm{d}x$.

Accepted Answer

$\frac{1}{2}\left[ -\frac{\pi }{2}+{{e}^{\frac{\pi }{2}}}-1 \right]$

2014 TPJC P1 Q4

2014 TPJC P1 Q4