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Math Help - Exponential and Trig in One Integral

  1. #1
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    Exponential and Trig in One Integral

    How do you integrate \int e^{x}\cos{(2x)}dx?

    I started to try integration by parts but I noticed quite quickly that it would not work because I would continue to get the same integral. Can anyone explain?
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  2. #2
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    Quote Originally Posted by Keithfert488 View Post
    How do you integrate \int e^{x}\cos{(2x)}dx?

    I started to try integration by parts but I noticed quite quickly that it would not work because I would continue to get the same integral. Can anyone explain?
    move that "repeated" integral to the side of the equation with the original integral.
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  3. #3
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    Quote Originally Posted by Keithfert488 View Post
    How do you integrate \int e^{x}\cos{(2x)}dx?

    I started to try integration by parts but I noticed quite quickly that it would not work because I would continue to get the same integral. <<<< That's the trick!
    Can anyone explain?
    Use integration by parts:

    First step:

    \int e^{x}\cos{(2x)}dx = e^x \cdot \cos(2x)+\int2e^x \cdot \sin(2x)dx

    Second step:

    \int e^{x}\cos{(2x)}dx = e^x \cdot \cos(2x)+2e^x \cdot \sin(2x)- \int 4e^x\cdot \cos(2x)dx

    Now collect the integrals at the LHS:

    5\int e^{x}\cos{(2x)}dx = e^x \cdot \cos(2x)+2e^x \cdot \sin(2x)

    Therefore:

    \int e^{x}\cos{(2x)}dx = \frac15 e^x \cdot (\cos(2x)+2 \sin(2x))

    EDIT: ...as usual: Too late
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  4. #4
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    Oh I see now. Thanks to both of you.
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