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Math Help - U Substitution with definite integral help

  1. #1
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    U Substitution with definite integral help

    Hi all,

    I have a homework problem (not graded, so I'm not coming here to boost my grade if that was anybody's concern :P) that I got caught up on. I'm not sure what to set u equal to, I originally though square root of x but that was getting extremely messy. I was hoping you guys could help walk me through this one to give me an idea of where I should go with it.

    Here's the problem:


    U Substitution with definite integral help-mathproblem.png


    Thanks guys!
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: U Substitution with definite integral help

    I would let:

    u=\sqrt{x} and since 1\le x\le4 we may state:

    u^2=x\,\therefore\,2u\,du=dx and we have:

    2\int_1^2 u^2-1\,du
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    Re: U Substitution with definite integral help

    Quote Originally Posted by MarkFL2 View Post
    I would let:

    u=\sqrt{x} and since 1\le x\le4 we may state:

    u^2=x\,\therefore\,2u\,du=dx and we have:

    2\int_1^2 u^2-1\,du
    Thanks so much! It didn't even cross my mind to set it equal to x.
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  4. #4
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    Re: U Substitution with definite integral help

    Equivalently, \frac{x- 1}{\sqrt{x}}= \frac{x}{x^{1/2}}- \frac{1}{x^{1/2}}= x^{1- 1/2}- x^{0- 1/2}= x^{1/2}- x^{-1/2}
    so the integral become \int_1^4 x^{1/2}- x^{-1/2}dx and you can integrate both parts by using \int x^ndx= \frac{x^{n+1}}{n+1}+ C.
    Thanks from MarkFL
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