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