1. ## Integrate

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

2. Substitute $\displaystyle x = e^u$ and then use integration by parts.

3. I am not very familiar with integrating (high school student) and this is a multiple choice(you can take the derivative of the answers) question how can I integrate partially?

4. ## Integration by parts

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 $\displaystyle e^x (x-1)$.