Question

Calculus integral question

Answer 1

what does it mean if there is a derivative in front of integral?

Answer 2

y do you think im confused :/

Answer 3

Does it happen to look like anything similar you've learned?
Like any particular rules or theorems?

Answer 4

i think i should take the integral, cause the top number is x^2, then i should the the derivative of the the solved integral, am i right?

Answer 5

ive never seen a derivative of an integral before

Answer 6

You could try. However, I don't think you'll have much luck trying to integrate e^(t^3) dt without some specialized tricks. I invite you to try, if you really want. :p
Instead, I think we want to use this one. It should be a lot more painless. This is considered the fundamental thm of calculus, first part.
\( \displaystyle \frac{d}{dx} \int_{a}^{x} f(t) \ dt = f(x) \)

Answer 7

It is quite close to what we have, with one exception in the bound. In the statement, the bounds are from a to x; a is a constant. In our problem, we seem to have an x^2 instead of x.

Answer 8

To deal with it, we'll need to find a way to differentiate the integral with respect to x^2 instead of x.
We'd find most luck using chain rule. Does this make sense so far?

Answer 9

Hm...
Let's say we switched out the integral for a simpler expression.
We'll call it \( F(x^2) \). F(u) has the property that \(\dfrac{d}{du} F(u) = e^{u^3} \)
We are taking the derivative of \(F(x^2) \) with respect to x. Symbolically...
\( \dfrac{d F(x^2)}{dx} \)
But our argument is x^2. We should use chain rule here to differentiate with respect to x^2 instead. Written:
\( \dfrac{dF(x^2)}{dx} = \dfrac{dF(x^2)}{d(x^2)} \times \dfrac{d(x^2)}{dx} \)
This makes it the derivative of F(x^2) with respect to x^2, and we can take the derivative of x^2 much more easily.