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Math Help - Using the Fundamental Theorem of Calculus...

  1. #1
    PTL
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    Using the Fundamental Theorem of Calculus...

    I'm asked to state the fundamental theorem of calculus, and use it to show that
    d/dx
    integral from x to tan x  \frac{1}{1+t^{2}}=\frac{x^{2}}{1+x^{2}}.

    I've been trying to use the wikipedia definition:
    Fundamental theorem of calculus - Wikipedia, the free encyclopedia

    So I have f(t)=1/(1+t^2)
    a = x, b= tan x

    But what do I use for the F(x)?

    Or is one supposed to use some corrollary?

    And to 'show something using the Fun. Theorem of Caculus', does one simply say: "Thus, by the Fundamental Theorem of Calculus.. qed."?
    Last edited by PTL; August 5th 2009 at 07:09 AM.
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  2. #2
    MHF Contributor red_dog's Avatar
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    F(x)=\arctan x
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  3. #3
    PTL
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    So that gives arctan(tan(x)) - arctan(x)?
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  4. #4
    Super Member Random Variable's Avatar
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    You don't need to know F(x) explicitly.

     \frac{d}{dx} \int^{\tan x}_{x} \frac{1}{1+t^{2}} \ dt

    let  F(x) = \int^{x}_{0} \frac{1}{1+t^{2}} \ dt

    then  \int^{\tan x}_{x} \frac{1}{1+t^{2}} \ dt = F(\tan x) - F(x)

     \frac{d}{dx} \int^{\tan x}_{x} \frac{1}{1+t^{2}} \ dt = F'( \tan x) \frac{d}{dx} (\tan x) - F'(x)

     = \frac {\sec^{2} x}{1+ \tan^{2}x} - \frac{1}{1+x^{2}}

     = 1 - \frac{1}{1+x^{2}} = \frac{x^{2}}{1+x^{2}}
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