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Thread: need some advice for writing out definite integrals!

  1. #1
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    need some advice for writing out definite integrals!

    So, i've been doing some definite integral problems that require various change of bounds, etc. It had just occured to me that i can easily forget to switch bounds, or get confused about which bounds im using and what not.

    What is the best way to do the change of bounds part for efficiency and neatness? (ex: changing bounds in terms of u before evaluating or after finding the integral?)

    (P.S: i trying to figure out what to do for my exam.... I want it to be acceptable too!)
    Last edited by lc99; Feb 25th 2018 at 08:42 PM.
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  2. #2
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    Re: need some advice for writing out definite integrals!

    As part of the substitution.

    Ex: $\displaystyle \int_{\tfrac{-1+\sqrt{5}}{2}}^{\tfrac{-1+\sqrt{1+4e}}{2}} \dfrac{(2x+1)dx}{x^2+x}$

    $u=x^2+x $
    $du=(2x+1)dx $
    $u\left( \dfrac{-1+\sqrt{5}}{2} \right) = 1$ (lower bound)
    $u\left( \dfrac{-1+\sqrt{1+4e}}{2} \right)=e $ (upper bound)

    $\displaystyle \int_1^e\dfrac{du}{u} = \ln e - \ln 1 = 1$
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    Re: need some advice for writing out definite integrals!

    oh okay, i see. So, you did the substitution before further integrating.
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  4. #4
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    Re: need some advice for writing out definite integrals!

    Quote Originally Posted by lc99 View Post
    oh okay, i see. So, you did the substitution before further integrating.
    Exactly. In general, you have:

    $\displaystyle \int_a^b f(u(x))u'(x)dx = \int_{u(a)}^{u(b)}f(u)du$
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