Answer
a. $F(x)=\displaystyle \frac{1}{2}x^{2}+2x$
b. see "step by step"
Work Step by Step
$\displaystyle \int kdx=kx+C, \quad \int x^{n}dx=\frac{x^{n+1}}{n+1}+C, n\neq-1 $
$\displaystyle \int[f(x)\pm g(x)]dx=\int f(x)dx\pm\int g(x)dx$
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a.
$\displaystyle \int_{0}^{x}(t+2)dt=[\frac{t^{2}}{2}+2t]_{0}^{x}=\frac{1}{2}x^{2}+2x$
b.
$\displaystyle \frac{d}{dx}[\frac{1}{2}x^{2}+2x]=\frac{1}{2}(2x^{2-1})+2(1)= x+2$
$f(x)=x+2\Rightarrow f(t)=t+2$
