f ′ ( x ) = ∫ cos ( x ) ln ( s i n ( x ) ) ⋅ d x {\displaystyle f'(x)=\int _{}^{}\cos(x)\ln(sin(x))\cdot dx}
∫ cos ( x ) ln ( sin ( x ) ) ⋅ d x = ∫ ln ( u ) ⋅ d u = u l n ( u ) − ∫ d u = u ⋅ l n ( u ) − u + c {\displaystyle \int _{}^{}\cos(x)\ln(\sin(x))\cdot dx=\int _{}^{}\ln(u)\cdot du~~~=~~~~uln(u)-\int _{}^{}du=u\cdot ln(u)-u+c} u = sin ( x ) z = ln ( u ) d w = d u {\displaystyle u=\sin(x)~~~~~~~~~~~~~~~~~~~~~~~~~~z=\ln(u)~~~~~dw=du} d u = cos ( x ) ⋅ d x d z = 1 x ⋅ d u w = u {\displaystyle du=\cos(x)\cdot dx~~~~~~~~~~~~~~~dz={\frac {1}{x}}\cdot du~~w=u} Therefore, f ( x ) = s i n ( x ) l n ( s i n ( x ) ) − s i n ( x ) + c {\displaystyle {\text{Therefore, }}f(x)=sin(x)ln(sin(x))-sin(x)+c}
