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Math Help - Two integrations

  1. #1
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    Two integrations

    Hi all,

    Got the good old exam tomorrow and I'm running through past papers. These two questions are giving me a few problems and unfortunately being multiple choice I don't have worked examples.

    Methods we've learnt basically sum up into substitution, partial fractions, t-substitution, integration by parts. If you could solve the problems using these methods it would be greatly appreciated.

    Here are the two.

     \int \cos(\ln(x)) dx

     \int \sin^2 x \cos 3x dx

    Thanks for your help guys.
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  2. #2
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    integrate sin(x)^2 *cos(3x) - Wolfram|Alpha
    wolframalpha will integrate for you and then you can click show steps in the right corner, and it will give you a step by step runthrough of how it got the answer
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  3. #3
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    \int \cos(\ln{x}) \, dx

    u = \ln{x}

    x = e^u

    dx = e^u \, du

    \int \cos{u} \cdot e^u \, du

    use parts
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  4. #4
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     \int \cos(\ln(x)) dx

    Try parts.

    Let u=cos(ln(x)), \;\ du=\frac{-sin(ln(x))}{x}dx, \;\ dv=dx, \;\ v=x

    We get:

    xcos(ln(x))+\int sin(ln(x))dx

    Now, do it again:

    u=sin(ln(x)), \;\ du=\frac{cos(ln(x))}{x}dx, \;\ v=x, \;\ dv=dx

    \int cos(ln(x))dx=xcos(ln(x))+xsin(ln(x))-\int cos(ln(x))dx

    See, we're going in circles. Now, add the integral on the right side to both sides:

    2\int cos(ln(x))dx=xcos(ln(x))+xsin(ln(x))

    Divide by 2:

    \int cos(ln(x))dx=\frac{xcos(ln(x))}{2}+\frac{xsin(ln(x  ))}{2}
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