Mathematics - Indefinite Integration Question with Solution | TestHub

$\mathrm{F}(\mathrm{x})=\int {\mathrm{e}}^{{\sin }^{-1}\mathrm{x}}\left(1-\frac{\mathrm{x}}{\sqrt{1-{\mathrm{x}}^2}}\right)\mathrm{dx}$

$$\begin{array}{rlrlrlrl}& {\sin }^{-1}x=t\Rightarrow x=\sin t\Rightarrow dx=\mathrm{costdt}& & & & & & \end{array}$$

$$\therefore F(x)=\int e^t\left(1-\frac{\sin t}{\cos t}\right)\cdot \cos tdt=\int e^t(\cos t-\sin t)dt=e^t\cos t+c$$

$$F(x)=e^{s^{-1}x}\cos \left({\sin }^{-1}x\right)+c$$

$$F(0)=1+c=1\Rightarrow c=0$$

$$\therefore F(x)=e^{{\sin }^{-1}x}\cos \left({\sin }^{-1}x\right)$$

$$F\left(\frac{1}{2}\right)=e^{\pi /6}\cos \frac{\pi }{6}=\frac{e^{\pi /6}\cdot \sqrt{3}}{2}=\frac{k\cdot \sqrt{3}{e}^{\pi /6}}{\pi }$$

$$\Rightarrow k=\frac{\pi }{2}$$