- #1

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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]

- Thread starter CantorSet
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- #1

- 44

- 0

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]

- #2

Hurkyl

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

with suitable conditions on the three functions.

- #3

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Well, here's my guest for the formula:ideaof 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.

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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- #5

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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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