g(x)=(x + 3)^2\ng(x)=g(x)\n∫_{-1}^{7}g(x)dx=□

Explanation:

Step1: Expand $G(x)$

$G(x)=(x + 3)^2=x^2+6x + 9$

Step2: Differentiate $G(x)$ to get $g(x)$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $g(x)=G'(x)=2x+6$

Step3: Integrate $g(x)$ from $-1$ to $7$

$\int_{-1}^{7}(2x + 6)dx=\int_{-1}^{7}2xdx+\int_{-1}^{7}6dx$. For $\int_{-1}^{7}2xdx$, using the power - rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we get $2\times\frac{x^{2}}{2}\big|{-1}^{7}=x^{2}\big|{-1}^{7}=7^{2}-(-1)^{2}=49 - 1=48$. For $\int_{-1}^{7}6dx=6x\big|{-1}^{7}=6\times(7-(-1))=6\times8 = 48$. Then $\int{-1}^{7}(2x + 6)dx=48+48=96$

Answer:

$96$