Answer
$5$
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 $F(x)$:
$\displaystyle \int(12x^{5}+5x^{4}-6x^{2}+4)dx=\frac{12x^{6}}{6}+\frac{5x^{5}}{5}-\frac{6x^{3}}{3}+4x+C$
$=2x^{6}+x^{5}-2x^{3}+4x+C=$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=2x^{6}+x^{5}-2x^{3}+4x$
Next, evaluate:
$F(1)=2+1-2+4=5$
$F(0)=0$
Finally, apply the theorem:
$\displaystyle \int_{0}^{1}(x^{2}+2)dx=F(1)-F(0)$
$=5-0$
$=5$
