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## Main Question or Discussion Point

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!