# OK easy question time!

illwerral
Hi folks! I've taken Calculus I and Calculus II, and I'm honestly not that bad at calculus but there's one thing I never quite got which really troubles me. How does one go about evaluating the derivative of an integral with a variable limit of integration?

Now, I realize that you're supposed to use the fundamental theorem of calculus, and that it somehow works out that, for example:

$$d/dx\int_{a}^{x} 2t dt = 2x$$

But when I do this, I actually do the integration then do the differentiation... I guess I'm not confident that just replacing t with x (in the example I gave) will work in general, like on a really bad integral like:

$$d/dx\int_{a}^{x} \sqrt{1+t^3}$$

Does it really equal $$\sqrt{1+x^3}$$? I can't actually expand it out to see for sure...

Does this question of mine even make sense or am I crazy? Thanks!

daudaudaudau
$$\frac{d}{dx}\int_a^xf(t)dt=\frac{d}{dx}(F(x)-F(a))=f(x)$$

Homework Helper
the only way I can make sense of your question is to translate it like this:

"I like to do things the hard way. How can I do it easily?"

Well, maybe not. That still doesn't make much sense! Have you considered going carefully over the proof of the fundamental theorem?

illwerral
Thanks for the replies folks, it's becoming more clear to me. I think I'll have to go over the proof of the fundamental theorem again after it's had time to sink in a bit, but I can solve problems now without feeling as if I'm pulling this out of a bag of tricks!