Question

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

Answer 1

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

Differentiating on both sides

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

$⟹\sqrt{1+\sin x}f\left(x\right)=\sqrt{1+\sin x}\cos x$

$⟹f\left(x\right)=\cos x$