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 {\displaystyle \int _{}^{}\cos(x)\ln(\sin(x))\cdot dx=\int _{}^{}\ln(u)\cdot du} u = sin ( x ) {\displaystyle u=\sin(x)} d u = cos ( x ) d x {\displaystyle du=\cos(x)dx} = u ⋅ l n ( u ) − u + c {\displaystyle =u\cdot ln(u)-u+c} 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}
