Hello everyone while I was studying integrals i came across with a question I could not think how to act. Here it is $\displaystyle \int(\cos(\ln(x))dx$ How to solve this step by step? Thanks
Integration by parts is the reverse of the product rule. Consider $\displaystyle \int x e^x dx$. To integrate by parts we need to follow the form: $\displaystyle \int u dv = uv - \int v du$. So let $\displaystyle u=x$ and $\displaystyle dv = e^x dx$. Integrating $\displaystyle u$ and differentiating $\displaystyle dv$ gets us $\displaystyle du=dx$ and $\displaystyle v=e^x$. So now plug in the parts, thus making $\displaystyle \int x e^x dx = x e^x - \int e^x dx$ with the answer being $\displaystyle e^x (x-1)$.
Follow the substitution Maddas suggested, and integrate!