How to integrate cos ln(x)

 

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