1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    FeDeX_LaTeX

    User Avatar
    Gold Member

    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

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Fundamental theorem of calculus
Loading...