Prove Integral f&g Defined in R: Find Derivative

[daniel_i_l]

Homework Statement

f is a function defined as: for all x>-1 f(x) = 1/((ln(x+1))^2 + 1) and for x=-1 f(x)=0.
a) Prove that $$F(x) = \int^{x^2 + 2x}_{0} f(t)dt$$ is defined in R and has a derivative there. Find the derivative.

b) g is defined as: for all x>-1 g(x) = f(x) and for x=-1 g(x) = -1.
Is $$G(x) = \int^{x^2 + 2x}_{0} g(t)dt$$ defined in R? Does it have a derivative there?

Homework Equations

The Attempt at a Solution

a)first I proved that f is continues for all x>=-1 Then since x^2+2x>0 for all x in R F(x) is defined and it's derivative is easy to get with the chain rule. I got:
F'(x) = (2x+2)/((ln(x^2 + 2x +1))^2 + 1)

b)Since g differs from f in only one point F(x) = G(x) and so G is defined and has a derivative for all x in R.

Is that right? I'm mostly worried about (b).
Thanks.

 

[Gib Z]

Looks good to me :) Other than the small error than G is defined and has a derivative for all x > -1, not all real values.

 

[HallsofIvy]

In (2), it helps to notice, as you said for (1), that x²+ 2x is never negative: the interval of integration, 0 to x²+ 2x, does not contain -1 and so what happens there is irrelevant.