If  $z\underset{0}{\overset{z}{\int }}\sin \left(f\left(x\right)\right)dx=\left(z+2\right)\underset{0}{\overset{z}{\int }}x\sin \left(f\left(x\right)\right)dx$ where z > 0, then  $\left|\left(1+z\right)f^{\mathbf{|}}\left(z\right)\cot \left(f\left(z\right)\right)\right|$ is equal to

 $\underset{0}{\overset{z}{\int }}\sin \left(f\left(x\right)\right)\left(1-x\right)dx=\left(z^2+z\right)\sin \left(f\left(z\right)\right)$
differentiate both sides w.r.t ‘z’
 $$\begin{gathered} \sin f\left(z\right)\left(1-z\right)=\left(2z+1\right)\sin \left(f\left(z\right)\right)+\left(z^2+z\right)\cos \left(f\left(z\right)\right)f^{\mathbf{|}}\left(z\right) \\ -3z\sin f\left(z\right)=\left(z^2+z\right)\cos \left(f\left(z\right)\right)f^{\mathbf{|}}\left(z\right) \\ -3=\left(z+1\right)\cot \left(f\left(z\right)\right).f{}^{|}\left(z\right) \end{gathered}$$