Ok I know the first fundamental theorem of Calc but this question is confusing me because both of the bounds are varibles with exponents. Please help

Find the derivative of https://webwork.math.ohio-state.edu/...5a7bc820c1.png

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- Jan 17th 2011, 10:54 AMrawkstar1st Fundamental Theorem of Calc
Ok I know the first fundamental theorem of Calc but this question is confusing me because both of the bounds are varibles with exponents. Please help

Find the derivative of https://webwork.math.ohio-state.edu/...5a7bc820c1.png - Jan 17th 2011, 11:07 AMFernandoRevilla
- Jan 18th 2011, 08:00 AMHallsofIvy
You can fix the problem that the lower limit of integration is a variable by breaking it into to integrals:

where "a" is any fixed number- the lower limit won't be relevant to the derivative.

To fix the problem that the upper limit is not just "x", make a change of variable. If we let , then when [tex]t= x^7[tex], u will be equal to x. , of course, and so the first integral becomes

.

For the second integral let so that when , . and so the integral becomes

That is,

and now you can easily apply the Fundamental Theorem of Calculus to the two integrals on the right. The derivative with respect to x is, modulo any careless errors,

In fact, that can be generalized to "Leibniz' formula":

which is, I suspect, how FernandoRevilla got his answer so quickly! - Jan 18th 2011, 08:33 AMFernandoRevilla
I got it in the following way:

Using the Fundamental Theorem of Calculus and Chain's Rule we inmediately obtain:

(of course differentiable, etc)

Fernando Revilla