- #1
illwerral
- 32
- 0
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:
[tex]
d/dx\int_{a}^{x} 2t dt = 2x
[/tex]
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:
[tex]
d/dx\int_{a}^{x} \sqrt{1+t^3}
[/tex]
Does it really equal [tex] \sqrt{1+x^3} [/tex]? I can't actually expand it out to see for sure...
Does this question of mine even make sense or am I crazy? Thanks!
Now, I realize that you're supposed to use the fundamental theorem of calculus, and that it somehow works out that, for example:
[tex]
d/dx\int_{a}^{x} 2t dt = 2x
[/tex]
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:
[tex]
d/dx\int_{a}^{x} \sqrt{1+t^3}
[/tex]
Does it really equal [tex] \sqrt{1+x^3} [/tex]? I can't actually expand it out to see for sure...
Does this question of mine even make sense or am I crazy? Thanks!