(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

f(x) is defined as f(x) = 1/((ln(x+1))^2 + 1) for all x>-1 and f(x)=0 for x=-1.

1)Prove that the function [tex]F(x) = \int^{x^2 + 2x}_{0} f(t)dt[/tex]

is defined and has a derivative in R.

2)g(x) is defined as g(x)=f(x) for x>-1 and g(x)=-1 for x=-1.

Also, [tex]G(x) = \int^{x^2 + 2x}_{0} g(t)dt[/tex]

Is G(x) defined in R? Does it have a derivative?

2. Relevant equations

3. The attempt at a solution

1) By taking the limit of f(x) at x=0 we see that f is continues for all x>=1 and since

x^2 + 2x >= -1 for all x in R F(x) is defined and is has a derivative from the chain rule.

2)Since f(x)=g(x) for all x=/=-1 F(x)=G(x) and so the answer to both questions is yes.

Are those right? I think that the answer to (2) is wrong but why?

Thanks.

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# Homework Help: Prove that the function in defined

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