Question

Differentiate the following integral with respect to x - please can someone explain verrryyy slowly? :(

Answer 1

\[\int\limits_{x}\sin(e^t)dt {}\]

Answer 2

This can be solved using the `Fundamental Theorem of Calculus: Part 1`

\(\large \dfrac{d}{dx}\int_c^x f(t)dt=f(x)\)

If that looks confusing, ignore it for now.
Let's go through this step by step.

Answer 3

The idea is, you're taking the integral of something.. then you're undoing the integration by taking it's derivative.

We care about what happens between those two steps though.
Remember that after you integrate, you evaluate the function at the limits.

So what's happening is:
~Integrating giving us something..
~plugging in the limits, giving us a function of x, instead of t..
~differentiating, giving us back the thing we started with, but now in terms of x, not t.

Should we do an easy example so it makes sense? :o

Answer 4

Yes please :P that makes a lot more sense already!

Answer 5

\[\large \frac{d}{dx}\int\limits_0^x 3\color{orangered}{t^2}dt\]

We perform the integration first,\[\large \frac{d}{dx}t^3|_0^x \qquad = \qquad \frac{d}{dx}\left[x^3-0^3\right]\]Now from here we can apply the derivative operation.
This will leave us with,\[\large \color{orangered}{3x^2}\]


See how we ended up with the same thing we started with, just in terms of x instead of t?
Make sure you're comparing the orange terms hehe.

Answer 6

So with your problem you just want to remember these little shortcuts.
You don't actually want to perform the integration OR differentiation.

Answer 7

With the integral you wrote, what is the upper boundary?
It looks like the lower limit is x?

Answer 8

Sorry yeah the other limit is 0 :) and i just worked that one through it makes perfect sense :D i've got a trickier one here so i'll give it a go now :D

Answer 9

thank you so much <3

Answer 10

kk c:

Answer 11

x^2 in the limit, using chain rule, gives 2xsin(e^x^2) . You're. A. Star.

Answer 12

Ooo good job! :)