# Differentiating integrals

1. Jan 30, 2009

### Appa

1. The problem statement, all variables and given/known data

$$d/dx$$ ($$\int$$$$^{x}_{0}$$ x2t2dt)
So the problem is to solve the derivative of the integral $$\int$$ x2t2dt from 0 to x.

2. Relevant equations

$$d/dx$$ ($$\int$$$$^{x}_{a}$$ f(t)dt) = f(x)

3. The attempt at a solution

I'm really unsure of how this should be computed but this was my guess:

$$d/dx$$ ($$\int$$$$^{x}_{0}$$ x2t2dt) = $$d/dx$$ ($$1/3$$x2(x)3 -($$1/3$$x2(0)3)) = $$d/dx$$ ($$1/3$$x5) = $$5/3$$x4

So, first I calculated the integral with respect to t and then derivated it with respect to x. But it feels wrong. I don't know how to treat the function x2t2 because the variable x is both part of the function and an endpoint of the interval for integration.

2. Jan 30, 2009

### Dick

You don't have to treat x^2 as 'part of the function'. It's just a constant multiplying the function t^2. You can take it out of the integral. If you have to do more complicated problems where the x and t are mixed up so you can't do that, check out the Leibniz Integral Rule. So your answer is correct.