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Thread: Integrate

  1. #1
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    Post 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
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  2. #2
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    Substitute $\displaystyle x = e^u$ and then use integration by parts.
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  3. #3
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    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?
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  4. #4
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    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 $\displaystyle e^x (x-1)$.

    Follow the substitution Maddas suggested, and integrate!
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  5. #5
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    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
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