Evaluating the Derivative of an Integral with Variable Limit

[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]

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

 

[HallsofIvy]

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!