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

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- Apr 28th 2010, 01:01 PMJohnDoeIntegrate
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

- Apr 28th 2010, 01:03 PMmaddas
Substitute $\displaystyle x = e^u$ and then use integration by parts.

- Apr 28th 2010, 01:08 PMJohnDoe
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?

- Apr 28th 2010, 01:40 PMNoobzUseRezIntegration 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 $\displaystyle e^x (x-1)$.

Follow the substitution Maddas suggested, and integrate! - Apr 28th 2010, 01:51 PMJohnDoe
I have integrated and found the solution but do I wonder if I have to substitute lnx with u by instict what if it was an another function? Thanks for your help BTW