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Homework Help: Fundamental theorem of calculus

  1. Nov 9, 2014 #1
    • Member warned about posting with no effort
    1. The problem statement, all variables and given/known data
    Using Fundamental Theorem of Calculus to find the derivative

    2. Relevant equations
    upper limit=x^2, lower limit=4x

    ∫ { 1 / [1+ (sin t)^2] }dt

    3. The attempt at a solution
    two independent variables are involved, how should i find the derivative?
  2. jcsd
  3. Nov 9, 2014 #2


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    Let [itex]F(x) = \int_{4x}^{x^2} \frac{dt}{1 + \sin^{2}t},[/itex] and put [itex]G(y) := \int_{0}^{y} \frac{dt}{1 + \sin^{2}t}.[/itex]

    Then, [itex]F(x) = G(x^2) - G(4x),[/itex] by the domain splitting property of the Riemann integral. Does this help things?
  4. Nov 9, 2014 #3
    oo) yup.... sure... i think i know how to solve now...... thanks
  5. Nov 9, 2014 #4


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    Not really. t is a 'dummy variable' that has no existence outside the integral. The integral as a whole is a function of x only.
  6. Nov 10, 2014 #5
    It's probably worth making an attempt using the first FTC and understanding it.

    You could use this,
    Set ##F(x) = \int_{\alpha(x)}^{\beta(x)}{ f(t) dt}##. Then,

    ##F'(x) = f(\beta(x))\beta'(x) - f(\alpha(x))\alpha'(x)##

    However it's probably worth understanding the first FTC before jumping into this.
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