Question

A function $f\left(x\right)$ satisfies $f\left(x\right)=\sin x+{\int }_0^x{f}^{\prime }\left(x\right)\left(2\sin t-{\sin }^{2}t\right)dt$. Find $\sqrt{f\left(x\right)}$

Answer 1

$f\left(x\right)=\sin x+{\int }_0^x{f}^{\prime }\left(x\right)\left(2\sin t-{\sin }^{2}t\right)dt$

Differentiating both sides, we get

$f^{\prime }\left(x\right)=\cos x+f^{\prime }\left(x\right).\left(2\sin x-{\sin }^{2}x\right)$

$f^{\prime }\left(x\right)\left[1-2\sin x+{\sin }^{2}x\right]=\cos x$

$f\left(x\right)={\int }_0^x\frac{\cos x}{{\left(\sin x-1\right)}^2}=\frac{1}{1-\sin x}-1$

$=\frac{\sin x}{1-\sin x}$

$\therefore \sqrt{f\left(x\right)}=\sqrt{\frac{\sin x}{1-\sin x}}$