Question

If $\int \sqrt{1+\sin x}\cdot f\left(x\right)dx=\frac{2}{3}{(1+\sin x)}^{3/2}+C$, then $f\left(x\right)$ is equal to

• A. $\cos x$
• B. $\sin x$
• C. $\tan x$
• D. $1$

Answer 1

The correct option is A $\cos x$

$\int \sqrt{1+\sin x}.f\left(x\right)dx=\frac{2}{3}{(1+\sin x)}^{3/2}+c$

On differentiating both sides, we get

$\sqrt{1+\sin x}.f\left(x\right)=\frac{2}{3}.\frac{3}{2}{(1+\sin x)}^{1/2}.\cos x+0\Rightarrow f\left(x\right)=\cos x$