# Derivative of the product of 2 definite integrals

1. Aug 4, 2012

### Dba18

1. The problem statement, all variables and given/known data

Find f'(x) for (integral from 0 to x of cos^5(t)dt)* (integral from x^2 to 1 of e^t^2 dt). No differentiation allowed in the answer
2. Relevant equations

3. The attempt at a solution. I used the product rule and integrated then differentiated the first term --> cos^5x* integral of e, etc. but I'm stumped on what to do with the these one term. Obviously I leave it as is for the first part of the product rule. So it's the second half where I'm looking for the differential of the integral of e^t^2

2. Aug 4, 2012

### LCKurtz

You don't have to integrate then differentiate. Use the fundamental theorem of calculus:$$\frac d {dx}\int_a^x f(t)\, dt = f(x)$$

3. Aug 4, 2012

### Dba18

I apologize for being so dense, but how does that apply to the stated answer for the quiz of cos^5(x) * integral with e + integral with cos^5(t) * (-2x)e^x^4 it's the answer for the last of the four terms in the result that's throwing me. To me, it seems the derivative of the integral of e^t^2 should just be e^x^2.

Thanks,

4. Aug 4, 2012

### LCKurtz

You have given me more information now, so I can see what your problem is. You need to use the chain rule:$$\frac d {dx}\int_{x^2}^1 e^{t^2}\ dt = -\frac d {dx}\int_{1}^{x^2} e^{t^2}\ dt$$
You plug in the $x^2$ for the $t$ in $e^{t^2}$ but you must multiply by the derivative of $x^2$ using the chain rule. Some versions of Leibnitz rule show this.

5. Aug 4, 2012

### Dba18

$$( \int {cos^t dt})$$ *($$\int{ e^t^2 dt) }$$. I am trying out the text editor I just found on the site to see if I can more clearly write oute the problem. So I'll post to see if this comes out ok

Last edited: Aug 4, 2012
6. Aug 4, 2012

### Dba18

Well my first attempts at using the editor aren't so good. I'm going to go take the laundry out of the dryer and sit and play with your answer for a While to make sure I know what's going on. And the reason you inverted the integral is because it went from a higher number to a lower one, right?

7. Aug 4, 2012

### eumyang

You're mixing non-LaTeX symbols with the LaTeX commands. Just use one set of tex tags.
What you want to type between the tex tags is this:
\left( \int_{0}^{x} \cos^5 t\ dt \right) \cdot \left( \int_{x^2}^{1} e^{t^2}\ dt \right)
Which gives you this:
$$\left( \int_{0}^{x} \cos^5 t\ dt \right) \cdot \left( \int_{x^2}^{1} e^{t^2}\ dt \right)$$

Last edited: Aug 4, 2012
8. Aug 4, 2012

### SammyS

Staff Emeritus
Dba18,

Here's how I go about remembering what to do with something like $\displaystyle \frac{d}{dx} \left( \int_{x^2}^{1} e^{t^2}\ dt \right)\ .$

Let G(t) be the anti derivative of $\displaystyle e^{t^2}\,,$ so that $\displaystyle G\,'(t)=e^{t^2}\,,$

Then $\displaystyle \int_{x^2}^{1} e^{t^2}\ dt = G(1)-G(x^2)\ .$

Therefore, $\displaystyle \frac{d}{dx} \left( \int_{x^2}^{1} e^{t^2}\ dt \right)= \frac{d}{dx}\left(G(1)-G(x^2)\right)=\underline{\ \ \ ?\ \ \ } \ .$