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Math Help - Integrate the Following:

  1. #1
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    Integrate the Following:

    Hello, and thanks for the help in advance...I have a final in this tomorrow and need to get this stuff down pat.

    I'm not sure how to go about integrating the following:

    x*arcsin(x)dx

    I know that substitution is always worth trying, but neither substitution nor integration by parts seems to work well here. Would anyone have any suggestions?
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by NBrunk View Post
    Hello, and thanks for the help in advance...I have a final in this tomorrow and need to get this stuff down pat.

    I'm not sure how to go about integrating the following:

    x*arcsin(x)dx

    I know that substitution is always worth trying, but neither substitution nor integration by parts seems to work well here. Would anyone have any suggestions?
    Use Integration by parts...

    let u = arcsin(x) and dv = x dx

    then du = \frac{1}{\sqrt{1-x^2}} and v = \frac{x^2}{2}

    Then perform: \int x \times arcsin(x)dx = uv - \int v .\mbox{du}


    You can check your work here
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  3. #3
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    Quote Originally Posted by NBrunk View Post
    Hello, and thanks for the help in advance...I have a final in this tomorrow and need to get this stuff down pat.

    I'm not sure how to go about integrating the following:

    x*arcsin(x)dx

    I know that substitution is always worth trying, but neither substitution nor integration by parts seems to work well here. Would anyone have any suggestions?
    using parts, I sense that you're probably having a problem integrating

    \frac{1}{2} \int \frac{x^2}{\sqrt{1-x^2}} \, dx



    if so, use a trig sub ... x = \sin{t}

    dx = \cos{t} \, dt

    \frac{1}{2} \int \frac{\sin^2{t}}{\sqrt{1-\sin^2{t}}} \cos{t} \, dt

    \frac{1}{2} \int \frac{\sin^2{t}}{\cos{t}} \cos{t} \, dt

    \frac{1}{2} \int \sin^2{t} \, dt

    \frac{1}{4} \int 1 - \cos(2t) \, dt

    \frac{t}{4} - \frac{\sin(2t)}{8}

    \frac{t}{4} - \frac{\sin{t}\cos{t}}{4}<br />

    back substitute ...

    \frac{\arcsin{x}}{4} - \frac{x \sqrt{1-x^2}}{4} + C<br />

    so, you final solution should be ...

    \frac{x^2}{2}\arcsin{x} - \frac{\arcsin{x}}{4} + \frac{x \sqrt{1-x^2}}{4} + C
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