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
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!
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