Answer
0
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 $F(x)$:
$\displaystyle \int(x^{3}-2x)dx=\frac{x^{4}}{4}-\frac{2x^{2}}{2}+C$
$=\displaystyle \frac{x^{4}}{4}-x^{2}+C$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=\displaystyle \frac{x^{4}}{4}-x^{2}$
Next, evaluate:
$F(2)=\displaystyle \frac{16}{4}-4=0$
$F(-2)=\displaystyle \frac{16}{4}-4=0$
Finally, apply the theorem:
$\displaystyle \int_{-2}^{2}(x^{3}-2x)dx=F(2)-F(-2)$
$=0-0 =0$
