Prove that the function in defined

[daniel_i_l]

Homework Statement

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 $$F(x) = \int^{x^2 + 2x}_{0} f(t)dt$$
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, $$G(x) = \int^{x^2 + 2x}_{0} g(t)dt$$
Is G(x) defined in R? Does it have a derivative?

Homework Equations

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.

 

[Dick]

Seems fine to me. For 2), why would you think changing the value of a function at a single point could change the integral? The set {-1} has measure 0.

 

[daniel_i_l]

Thanks a lot.