# Integrate

• April 28th 2010, 01:01 PM
JohnDoe
Integrate
Hello everyone while I was studying integrals i came across with a question I could not think how to act. Here it is $\int(\cos(\ln(x))dx$ How to solve this step by step? Thanks
• April 28th 2010, 01:03 PM
Substitute $x = e^u$ and then use integration by parts.
Integration by parts is the reverse of the product rule. Consider $\int x e^x dx$. To integrate by parts we need to follow the form: $\int u dv = uv - \int v du$. So let $u=x$ and $dv = e^x dx$. Integrating $u$ and differentiating $dv$ gets us $du=dx$ and $v=e^x$. So now plug in the parts, thus making $\int x e^x dx = x e^x - \int e^x dx$ with the answer being $e^x (x-1)$.