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Thread: a couple integrals

  1. #1
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    a couple integrals

    1.
    $\displaystyle
    \int \frac{sinx}{cos^3x} dx
    $

    2. $\displaystyle \int x \sqrt {1-x} dx$

    3. if F(x)= $\displaystyle \int_{0}^{\sqrt x} \sqrt {t^2+20} dx
    $ find F'(16)

    Hi, any help would be wonderful. I really need the explanation more than the answer, though.
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  2. #2
    Super Member PaulRS's Avatar
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    1. Subsitute $\displaystyle u=\tan(x)$ remember that $\displaystyle u'=\frac{1}{cos^2(x)}$

    2. Subsitute $\displaystyle u=\sqrt[]{1-x}$

    3. Define $\displaystyle G(x)=\int_0^{x}{t^2+20}dt$ so that $\displaystyle G(\sqrt[]{x})=F(x)$(1)
    Now differentiate in (1) using the chain rule and remember that $\displaystyle G'(x)=x^2+20$
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  3. #3
    MHF Contributor
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    What are your thoughts?

    1. A substitution looks easy enough. u = ???

    2. Begging for Integration by Parts if you want to learn something, but a simple substitution of everything under the radical might be a lot easier.

    3. A derivative of an integral? Don't forget the Chain Rule. This one likely is ill-formed, Do you mean "dt"?

    Let's see what you get.
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  4. #4
    Moo
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    Hello,

    For the first one, a substitution u=cos(x) is quite a valid way to solve it too ^^
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  5. #5
    GAMMA Mathematics
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    Quote Originally Posted by Moo View Post
    Hello,

    For the first one, a substitution u=cos(x) is quite a valid way to solve it too ^^
    I would definitely follow this substitution. Follow PaulRS' advice on the 2nd integral.
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