Answer
$-\displaystyle \frac{15}{2}$
Work Step by Step
FTC: Let $f$ be a continuous function defined on the interval $[a, b]$,
and let $F$ be any antiderivative 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 antiderivative $F(x)$:
$\displaystyle \int(x-2)dx=\frac{x^{2}}{2}-2x+C$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=\displaystyle \frac{x^{2}}{2}-2x$
Next, evaluate:
$F(1)=\displaystyle \frac{1^{2}}{2}-2(1)=-\frac{3}{2}$
$F(-2)=\displaystyle \frac{(-2)^{2}}{2}-2(-2)=2+4=6$
Finally, apply the theorem:
$\displaystyle \int_{-1}^{1}(x^{2}+2)dx=F(1)-F(-1)$
$=-\displaystyle \frac{3}{2}-6=-\frac{15}{2}$
