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I am having trouble with the following question.

If $F(x)=\int_{0}^x xf(t) dt$, find F'(x).

Please help

Thank you in advance

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    Do you know about the Fundamental Theorem of Calculus?2012-10-25

1 Answers 1

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Well, notice first that you can pull $x$ out of the integral. Then, you can use the product rule and the fundamental theorem of calculus to obtain the answer. Remember that $$\frac{\text{d}}{\text{d}x}\int_{0}^x f(t)\text{d}t=f(x).$$

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    I get $F'(x)=\int_{0}^x f(t)dt +x(f(x)-f(0))$2012-10-25
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    Why is f(0)=0 ?2012-10-25
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    @user43418 let's say that we have $G(t)=\int f(t) dt$, so $\int_0^x f(t) dt=G(x)-G(0)$. When you take the derivative the constant term drops out, and you're left with $G'(x)=f(x)$.2012-10-25
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    So $F'(x)= \int_{0}^{x}f(t)dt+xf(x)$2012-10-25
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    Is that correct ?2012-10-25