7.1 Integration By Parts/29

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

$\int _{}^{}\cos(x)\ln(\sin(x))\cdot dx=\int _{}^{}\ln(u)\cdot du~~~=~~~~u\ln(u)-\int _{}^{}du=u\ln(u)-u+c$
$u=\sin(x)~~~~~~~~~~~~~~~~~~~~~~~~~~z=\ln(u)~~~~~dw=du$
$du=\cos(x)\cdot dx~~~~~~~~~~~~~~~dz={\frac {1}{x}}\cdot du~~w=u$

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

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