## Differentiation of Integral

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.
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 Recognitions: Homework Help 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.
 Recognitions: Gold Member Homework Help Science Advisor Differentiate it, don't try to integrate it!

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Homework Help

## Differentiation of Integral

I think he needs to evaluate the integral to be able to do that doesn't he >.<
 Recognitions: Gold Member Homework Help Science Advisor 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..

Mentor
 Quote by Gib Z I think he needs to evaluate the integral to be able to do that doesn't he >.<
No. The integrand does not involve x. Simply apply the fundamental theorem of calculus.

Hint:
$$\frac{d}{dx}\int_{x^3}^{e^x}\cos t^2dt = \frac{d}{dx}\int_0^{e^x}\cos t^2dt \;\;-\;\; \frac{d}{dx}\int_0^{x^3}\cos t^2dt$$
 So tha answer is $$2e^xsine^{2x}-6x^5sinx^6$$ yes?
 Mentor No. Read arildno's post again. Sine is not involved.
 ok ok,my mistake $$2e^{2x}cos(e^{2x})-6x^5cos(x^6)$$
 Recognitions: Gold Member Homework Help Science Advisor Eeh?? Where do you get that 2-factor from??
 if we put $$e^x$$ to t shouldn't it be $$e^{2x}$$
 Recognitions: Gold Member Homework Help Science Advisor I'm talking about the 2-factors in front of the cosine's, not the ones within the arguments.
 I typed wrongly instead of $$e^{2x}$$,I typed $$e^{x}$$ $$\frac{d}{dx}(e^{2x})=2xe^{2x}$$ This 2 are you asking ?
 Recognitions: Gold Member Homework Help Science Advisor What is a(x), and what is b(x); what are their derivatives?
 You say that answer is $$e^xcos(e^{2x})-3x^2cos(x^6)$$?
 Recognitions: Homework Help Today was not my best day obviously =] Yes I should have seen the proper method arildno and DH, maybe Ill have better luck tomorrow.