Find \(\displaystyle\displaystyle \displaystyle\int{\frac{{{\mathrm{e}}^{{{\sin }^{-1}}x}}}{\sqrt{1-

Question

Find $\displaystyle\displaystyle \displaystyle\int{\frac{{{\mathrm{e}}^{{{\sin }^{-1}}x}}}{\sqrt{1-{{x}^{2}}}}\mathrm{d}x}$[2] Express $\mathrm{f}\left( x \right)=\frac{{{x}^{2}}-6x+1}{\left( 3x+1 \right)\left( {{x}^{2}}+3 \right)}$ in the form $\frac{A}{3x+1}+\frac{Bx+C}{{{x}^{2}}+3}$, where the values of $A$, $B$ and $C$ to be determined. Hence, find $\displaystyle\displaystyle \displaystyle\int{\mathrm{f}\left( x \right)\,\mathrm{d}x}$.[2] Find the exact value of $\displaystyle\displaystyle \displaystyle\int_{\pi }^{2\pi }{{{\sin }^{2}}\left( \frac{x}{8} \right)\,\,\mathrm{d}x}$.[3]

Accepted Answer

(a) $e^{\sin^{-1}x} + C$ (b) $A=1$, $B=0$, $C=-2$, $\int{\text{f}\left( x \right)\,\text{d}x}=\frac{1}{3}\ln \left| 3x+1 \right|-\frac{2}{\sqrt{3}}{{\tan }^{-1}}\frac{x}{\sqrt{3}}+c$ (c) $\frac{\pi}{2} - 2 + \sqrt{2}$

2021 JPJC Promo Q10

2021 JPJC Promo Q10

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