How to integrate sin ln(x)

 

$$\int \sin lnx\mathrm{d}x$$ $$\begin{array}{l}BySubstitution\\ suppose\Rightarrow z=\ln x\\ e^z=x\\ dz=1/xdx\\ \int {\mathrm{e}}^{\mathrm{z}}\sin z\mathrm{d}z\\ ByParts\\ Let\Rightarrow u=\sin zdv=e^z\\ du=coszv=e^z\\ \int e^z\sin zdz=e^z\cdot \sin z-\int e^z\cos z\mathrm{d}z\\ ByParts\\ Let\Rightarrow g=\cos zdf=e^z\\ dg=-sinzf=e^z\\ \int e^{\mathrm{z}}\sin \mathrm{z}\mathrm{d}z=e^z\sin z-\left(e^z\cos z+\int e^z\sin z\mathrm{d}z\right)\\ \int {\mathrm{e}}^z\sin z\mathrm{d}z={\mathrm{e}}^z\sin z-{\mathrm{e}}^z\cos z\\ \int {\mathrm{e}}^z\sin z\mathrm{d}z=\frac{1}{2}\left(e^z\sin z-e^z\cos z\right)\end{array}$$