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Math Help - please need help-urgent

  1. #1
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    please need help-urgent

    Integrate sqrt(x)csc^2[x^(3/2)]dx
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by bobby77
    Integrate sqrt(x)csc^2[x^(3/2)]dx
    \int dx \, \sqrt{x} \, csc^2 \left ( x^{3/2} \right )

    Let y = x^{3/2}
    Thus dy = \frac{3}{2} \sqrt{x}\, dx

    \int dx \, \sqrt{x} \, csc^2 \left ( x^{3/2} \right ) = \frac{2}{3} \int dy \, csc^2(y)

     = -\frac{2}{3}cot(y) = -\frac{2}{3} cot \left ( x^{3/2} \right )

    -Dan
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by bobby77
    Integrate sqrt(x)csc^2[x^(3/2)]dx

    \frac{d}{dx} \left(\frac{2}{3} x^{3/2}\right) =x^{1/2}, so the
    integrand is the derivative of a multiple of \cot(x^{3/2}) .

    From here the fundamental theorem of Calculus should give you the answer.

    RonL
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by topsquark
    \int dx \, \sqrt{x} \, csc^2 \left ( x^{3/2} \right )

    Let y = x^{3/2}
    Thus dy = \frac{3}{2} \sqrt{x}\, dx

    \int dx \, \sqrt{x} \, csc^2 \left ( x^{3/2} \right ) = \frac{2}{3} \int dy \, csc^2(y)

     = -\frac{2}{3}cot(y) = -\frac{2}{3} cot \left ( x^{3/2} \right )

    -Dan
    Actually the answer is:  -\frac{2}{3} cot \left ( x^{3/2} \right ) + C. Don't get sloppy like me and forget the constant!

    -Dan
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