If $f(x)=\int e^{\sin x}\left(\frac{5 \cos x+\cos ^3 x}{(2-\sin x) \sqrt{3+\cos ^2 x}}\right) d x \q

Question

If $f(x)=\int e^{\sin x}\left(\frac{5 \cos x+\cos ^3 x}{(2-\sin x) \sqrt{3+\cos ^2 x}}\right) d x \quad$ and $f(0)=1 \quad$ then value of $f(\pi / 2) \quad$ equals

Professor PC Thomas Classes · Answer

$$
\int e^{\sin x}\left(\frac{5+\cos ^2 x}{(2-\sin x) \sqrt{3+\cos ^2 x}}ight) \cos x d x
$$
put $\sin x=t$
$$
\begin{aligned}
& \int e^t\left(\frac{6-t^2}{(2-t) \sqrt{4-t^2}}ight) d t \
& \int e^t\left(\frac{4-t^2}{(2-t) \sqrt{4-t^2}}+\frac{2}{(2-t) \sqrt{4-t^2}}ight) d t \
& \int e^t\left(\sqrt{\frac{2+t}{2-t}}+\frac{2}{(2-t)^{3 / 2}(2+t)^{1 / 2}}ight) d t
\end{aligned}
$$
If $g(t)=\sqrt{\frac{2+t}{2-t}}$
$$
\begin{aligned}
& g^{\prime}(t)=\frac{2}{(2-t)^{3 / 2}\left(2+t^2ight)^{1 / 2}} \
& e^t \sqrt{\frac{2+t}{2-t}}+c
\end{aligned}
$$
$$
\begin{aligned}
& f(x)=e^{\sin x} \sqrt{\frac{2+\sin x}{2-\sin x}+c} \
& f(0)=1 \Rightarrow c=0 \
& f(x)=e^{\sin x} \sqrt{\frac{2+\sin x}{2-\sin x}} \
& f\left(\frac{\pi}{2}ight)=\sqrt{3} e
\end{aligned}
$$