Answer
$\approx-0.904545$
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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$f(x)=2.1x-4.3x^{1.2}$, which is continuous on [$0,1$].
First, find an antiderivative $F(x)$:
$\displaystyle \int f(x)dx=\frac{2.1x^{2}}{2}-\frac{4.1x^{2.2}}{2.2}+C$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=\displaystyle \frac{2.1x^{2}}{2}-\frac{4.1x^{2.2}}{2.2}$
Next, evaluate:
$ F(1)=\displaystyle \frac{2.1(1)}{2}-\frac{4.3(1)}{2.2}\approx$-0.904545454545
$F(0)=0$
Finally, apply the theorem:
$\displaystyle \int_{0}^{1}f(x)dx=F(1)-F(2)\approx-0.904545$
