A calculus problem

We know that: $g$ is the derivative of $G$ , which means $G$ is an antiderivative of $g$ .

• Since we know the antiderivative of $g$ , we can use the fundamental theorem of calculus: For every function $g$ and its antiderivative $G$ , $\displaystyle\int_{a}^{b}$ $g(x)dx=$ $G(b)-G(a).$ $\displaystyle\int_{-1}^{7}$ $g(x)dx=$ $G(7)-G(-1)=$ $(7+3^{2}-(-1)+3^{2}=$ $100-4=$ $96$

$\displaystyle I = \int_{-1}^7 g(x)\ dx = \int_{-1}^7 G'(x)\ dx = G(7)-G(-1) = (7+3)^2 - (-1+3)^2 = 10^2-2^2 = \boxed{96}$ .