If x ∫_0^x (f(t)) d t=(x+2) ∫_0^x t (f(t)) d t where x>0 , then...

Differentiate both sides \begin{aligned} & x \sin (f(x))+\int_0^x \sin (f(t)) d t=(x+2) \sin (f(x))+\int_0^x t \sin (f(t)) d t \\ & \therefore \quad x^2 \sin (f(x))+x \sin (f(x))=\int_0^x \sin (f(t)) d t-\int_0^x t \sin (f(t)) d t \\ & \end{aligned} Again differentiate \begin{aligned} & x^2 \cos (f(x)) f^{\prime}(x)+2 x \sin (f(x))+\sin (f(x))+x \cos (f(x)) f^{\prime}(x)=\sin (f(x))-x \sin (f(x)) \\ & =x(x+1) \cos (f(x))+3 x \sin (f(x))=0 \\ & \Rightarrow \quad f^{\prime}(x) \cot (f(x))+\frac{3}{1+x}=0 \end{aligned}