7.1 Integration By Parts/29

$f'(x)=\int _{}^{}\cos(x)ln(sin(x))\cdot dx$

$\int _{}^{}\cos(x)ln(sin(x)\cdot dx=\int _{}^{}\ln(u)\cdot du$
$u=sin(x)$
$du=cos(x)dx$

$=u\cdot ln(u)-u+c$

${\text{Therefore, }}f(x)=sin(x)ln(sin(x))-sin(x)+c$