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Math Help - Integration with change of variable

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    Integration with change of variable

    Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?


    \int cos(x)(1+sin^2(x)) dx

    Thanks again!
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    Quote Originally Posted by aceband View Post
    Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?


    \int cos(x)(1+sin^2(x)) dx

    Thanks again!
    Substitute u = \sin x to get \int 1 + u^2 \, du ....
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    Quote Originally Posted by aceband View Post
    Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?


    \int cos(x)(1+sin^2(x)) dx

    Thanks again!
    You're on the right track.

    Note that if you let u = \sin{x} then \frac{du}{dx} = \cos{x}.

    So the integral becomes

    \int{\cos{x}(1 + \sin^2{x})\,dx} = \int{(1 + u^2)\,\frac{du}{dx}\,dx}

     = \int{1 + u^2\,du}

     = u + \frac{1}{3}u^3 + C

     = \sin{x} + \frac{1}{3}\sin^3{x} + C.
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