Answer
$\displaystyle \frac{14}{3}$
Work Step by Step
FTC: Let $f$ be a continuous function defined on the interval $[a, b]$,
and let $F$ be any anti-derivative of f defined on $[a, b]$. Then
$\displaystyle \int_{a}^{b}f(x)dx=\left[F(x)\right]_{a}^{b}=F(b)-F(a)$.
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First, find an anti-derivative $F(x)$,
$\displaystyle \int(x^{2}+2)dx=\frac{x^{3}}{3}+2x+C$
(we take C=0, since by FTC we need ANY anti-derivative)
$F(x)=\displaystyle \frac{x^{3}}{3}+2x$
Next, evaluate
$F(1)=\displaystyle \frac{1}{3}+2(1)=\frac{7}{3}$
$F(-1)=-\displaystyle \frac{1}{3}-2=-\frac{7}{3}$
Apply the theorem:
$\displaystyle \int_{-1}^{1}(x^{2}+2)dx=F(1)-F(-1)$
$=\displaystyle \frac{7}{3}-(-\frac{7}{3})=\frac{14}{3}$
