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 $F(x)$:
$\displaystyle \int(2x^{3}+x)dx=\frac{2x^{4}}{4}+\frac{x^{2}}{2}+C$
$=\displaystyle \frac{x^{4}}{2}+\frac{x^{2}}{2}+C=$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=\displaystyle \frac{x^{4}}{4}-x^{2}$
Next, evaluate:
$F(1)=\displaystyle \frac{1}{4}-1=-\frac{3}{4}$
$F(-1)=\displaystyle \frac{1}{4}-1=-\frac{3}{4}$
Finally, apply the theorem:
$\displaystyle \int_{-1}^{1}(2x^{3}+x)dx=F(1)-F(-1)$
$=-\displaystyle \frac{3}{4}-(-\frac{3}{4})=0$
