The fundamental theorem of calculus

  • Thread starter CantorSet
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  • #1
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Can we apply the fundamental theorem of calculus to an integrand that's a function of the differentiating variable?

For example, can one still use the fundamental theorem of calculus when trying to find the derivative of this function:

[tex]f(x) = \int_{0}^{x} \frac{cos(xt)}{t}dt[/tex]
 

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  • #2
Hurkyl
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Nope. But the idea of the proof can still be used -- you can derive a general formula for
[tex]\int_{f(x)}^{g(x)} h(x, t) \, dt[/tex]
with suitable conditions on the three functions.
 
  • #3
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Nope. But the idea of the proof can still be used -- you can derive a general formula for
[tex]\int_{f(x)}^{g(x)} h(x, t) \, dt[/tex]
with suitable conditions on the three functions.
Well, here's my guest for the formula:

If

[tex]H(x) = \int_{f(x)}^{g(x)} h(x, t) \, dt[/tex],

then

[tex]H'(x) = h(x,g(x)) - h(x,f(x))[/tex]
 
  • #4
Hurkyl
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Have you tried not guessing, and actually trying to work it out, starting with the very definition of derivative?
 
  • #5
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Have you tried not guessing, and actually trying to work it out, starting with the very definition of derivative?
Let

[tex]F(x) = \int_{a}^{g(x)} f(x,t) dt[/tex]


Now define

[tex]I(x,y) = \int f(x,t) dt [/tex]

with t = y after indefinite integration over t.

Then

[tex] F'(x) = I_x(x,g(x)) - I_x(x,a) + I_x(x,g(x))g'(x)[/tex]
 

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