Rewrite integral

Question

Need help to rewrite this integral.

$I=\int^{\pi/4}_{0}Cos(x)ln(x)dx$

Rewrite the integral $I$ in the form: $I=a-J$ with $J=\int^{\pi/4}_{0}h(x)dx$

$h(x)$ is continuous and bounded around zero.

Answer 1

$$I=\int^{\pi/4}_{0}Cos(x)ln(x)dx=\int^{x=\pi/4}_{x=0}ln(x)d\sin x$$$$=\sin x\ln x|_{x=0}^{x=\frac{\pi}{4}}-\int^{x=\pi/4}_{x=0}\sin(x)d ln(x)$$$$=\frac{\sqrt 2}{2}\ln(\frac{\pi}{4})-\int^{\pi/4}_{0}\frac{\sin x}{x}dx$$