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 ln ( u ) − ∫ d u = u ln ( u ) − u + c {\displaystyle \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 ) d w = d u {\displaystyle u=\sin(x)\qquad \qquad z=\ln(u)\qquad dw=du} d u = cos ( x ) ⋅ d x d z = 1 x ⋅ d u w = u {\displaystyle du=\cos(x)\cdot dx\qquad dz={\frac {1}{x}}\cdot du\qquad w=u} Therefore, f ( x ) = sin ( x ) ln ( sin ( x ) ) − sin ( x ) + c {\displaystyle {\text{Therefore, }}f(x)=\sin(x)\ln(\sin(x))-\sin(x)+c}
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