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Math Help - Integrate

  1. #1
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    Integrate

    The bottom limit is beta and the top limit is y the function is 1/x^(alpha +1) dx
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  2. #2
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    \int_\beta^y \frac{1}{x^{\alpha+1}} dx = \int_\beta^y x^{-\alpha-1} dx = -\frac{x^{-\alpha}}{\alpha+1}  |_{x=\beta}^y = -\frac{y^{-\alpha}}{\alpha+1}+\frac{\beta^{-\alpha}}{\alpha+1}= \frac{\beta^{-\alpha}-y^{-\alpha}}{\alpha+1}= \frac{y^\alpha-\beta^\alpha}{(\alpha+1)(y\beta)^\alpha}
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  3. #3
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    Hello;
    Quote Originally Posted by Media_Man View Post
    \int_\beta^y \frac{1}{x^{\alpha+1}} dx = \int_\beta^y x^{-\alpha-1} dx = -\frac{x^{-\alpha}}{\alpha+1}  |_{x=\beta}^y = -\frac{y^{-\alpha}}{\alpha+1}+\frac{\beta^{-\alpha}}{\alpha+1}= \frac{\beta^{-\alpha}-y^{-\alpha}}{\alpha+1}= \frac{y^\alpha-\beta^\alpha}{(\alpha+1)(y\beta)^\alpha}
    Hmmm except that an antiderivative of x^{-\alpha-1} is -\frac{x^{-\alpha}}{\alpha}, and not -\frac{x^{-\alpha}}{\alpha+1}
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  4. #4
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    Right you are

    Quote Originally Posted by Media_Man View Post
    \int_\beta^y \frac{1}{x^{\alpha+1}} dx = \int_\beta^y x^{-\alpha-1} dx = -\frac{x^{-\alpha}}{\alpha}  |_{x=\beta}^y = -\frac{y^{-\alpha}}{\alpha}+\frac{\beta^{-\alpha}}{\alpha}= \frac{\beta^{-\alpha}-y^{-\alpha}}{\alpha}= \frac{y^\alpha-\beta^\alpha}{\alpha(y\beta)^\alpha}
    *Yes, my bad. Quick fix, though.
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  5. #5
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    uhh that makes more sense. Thanks to both of you
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