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Math Help - Quick Integration Question

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    Quick Integration Question

    Let's say you had the integration problem (5x+7)^20. You would let u=5x+7 and du=5dx and so .2du=dx, and so you would end up with the integration of u^20 times .2 du. You could then bring out the constant term .2, and you would end up with .2 times the integration of u^20 du.

    However, I remember that there are times in which you would multiply one side of the equation by a constant, then multiply the other side by its reciprocal after integrating the terms inside the integral. When is this exactly done? I don't think you do this in u-substitution, but I distinctly remember some instances in which you multiply by the reciprocal, not by the normal constant.
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    Quote Originally Posted by jaijay32 View Post
    Let's say you had the integration problem (5x+7)^20. You would let u=5x+7 and du=5dx and so .2du=dx, and so you would end up with the integration of u^20 times .2 du. You could then bring out the constant term .2, and you would end up with .2 times the integration of u^20 du.

    However, I remember that there are times in which you would multiply one side of the equation by a constant, then multiply the other side by its reciprocal after integrating the terms inside the integral. When is this exactly done? I don't think you do this in u-substitution, but I distinctly remember some instances in which you multiply by the reciprocal, not by the normal constant.
    \int (5x+7)^{20} \, dx

    u = 5x+7

    du = 5 \, dx

    \frac{1}{5} \int (5x+7)^{20} \cdot 5 \, dx

    substitute ...

    \frac{1}{5} \int u^{20} \, du

    \frac{1}{5} \cdot \frac{1}{21} u^{21} + C

    \frac{1}{105} (5x+7)^{21} + C

    now ... where is your confusion?
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