# Differentiation of Integral

1. Oct 6, 2007

### azatkgz

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

$$\frac{d}{dx}\int_{x^3}^{e^x}cost^2dt$$

3. The attempt at a solution

$$\int cost^2dt=\frac{sint^2}{2t}+\int\frac{sint^2}{2t^2}dt$$
$$\int\frac{sint^2}{2t^2}dt=-\frac{sint^2}{2t}+\int cost^2dt$$
I came back to initial integral.

2. Oct 6, 2007

### Gib Z

That can happen with some of your choice for u and dv whilst doing integration by parts, the second time you apply it use different choices.

3. Oct 6, 2007

### arildno

Differentiate it, don't try to integrate it!

4. Oct 6, 2007

### Gib Z

I think he needs to evaluate the integral to be able to do that doesn't he >.<

5. Oct 6, 2007

### arildno

Nope.
Here's how to do it properly:
Let F(t) be an antiderivative of f, F'(t)=f(t).
Thus, we have:
$$\frac{d}{dx}\int_{a(x)}^{b(x)}f(t)dt=\frac{d}{dx}(F(b(x))-F(a(x)))=F'(b(x))b'(x)-F'(a(x))a'(x)=f(b(x))b'(x)-f(a(x))a'(x)$$

As you can see, you do not need the explicit form of F, only the guarantee that some such F exists..

6. Oct 6, 2007

### D H

Staff Emeritus
No. The integrand does not involve x. Simply apply the fundamental theorem of calculus.

Hint:
$$\frac{d}{dx}\int_{x^3}^{e^x}\cos t^2dt = \frac{d}{dx}\int_0^{e^x}\cos t^2dt \;\;-\;\; \frac{d}{dx}\int_0^{x^3}\cos t^2dt$$

Last edited: Oct 6, 2007
7. Oct 6, 2007

### azatkgz

So tha answer is
$$2e^xsine^{2x}-6x^5sinx^6$$
yes?

8. Oct 6, 2007

### D H

Staff Emeritus
No. Read arildno's post again. Sine is not involved.

9. Oct 6, 2007

### azatkgz

ok ok,my mistake
$$2e^{2x}cos(e^{2x})-6x^5cos(x^6)$$

Last edited: Oct 6, 2007
10. Oct 6, 2007

### arildno

Eeh??
Where do you get that 2-factor from??

11. Oct 6, 2007

### azatkgz

if we put $$e^x$$ to t shouldn't it be $$e^{2x}$$

12. Oct 6, 2007

### arildno

I'm talking about the 2-factors in front of the cosine's, not the ones within the arguments.

13. Oct 6, 2007

### azatkgz

I typed wrongly instead of $$e^{2x}$$,I typed $$e^{x}$$
$$\frac{d}{dx}(e^{2x})=2xe^{2x}$$
This 2 are you asking ?

14. Oct 6, 2007

### arildno

What is a(x), and what is b(x); what are their derivatives?

15. Oct 6, 2007

### azatkgz

You say that answer is
$$e^xcos(e^{2x})-3x^2cos(x^6)$$?

16. Oct 6, 2007

### Gib Z

Today was not my best day obviously =] Yes I should have seen the proper method arildno and DH, maybe Ill have better luck tomorrow.

17. Oct 6, 2007

### HallsofIvy

Staff Emeritus
No, no one has said that! Several people have asked you questions about this problem that you haven't answered.

18. Oct 6, 2007

### arildno

I didn't say that; but you said it correctly now!

19. Oct 6, 2007

### azatkgz

Ok.Thanks to everyone.

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