Evaluate the integral.\\
$\int_0^1 \left(2 + \frac{2}{5}u^4 - \frac{5}{4}u^9\right) du$
Part 1 of 4
An anti-derivative of $kx^n$, as long as $n \ne -1$, is $k\frac{x^{n+1}}{n+1}$
Part 2 of 4
Therefore, an antiderivative of $f(u) = 2 + \frac{2}{5}u^4 - \frac{5}{4}u^9$ is
$F(u) = 2u + \frac{2}{5}\frac{u^5}{5} - \frac{5}{4}\frac{u^{10}}{10}$
Part 3 of 4
Since $\left(\frac{2}{5}\right)\left(\frac{1}{5}\right) = \frac{4}{50}$ and $\left(\frac{5}{4}\right)\left(\frac{1}{10}\right) = \frac{50}{40}$, we now have
$\int_0^1 \left(2 + \frac{2}{5}u^4 - \frac{5}{4}u^9\right) du = \left[2u + \frac{2}{25}u^5 - \frac{1}{8}u^{10}\right]_0^1$
To evaluate, we substitute 1 and 0 into the antiderivative $F(u) = 2u + \frac{2}{25}u^5 - \frac{1}{8}u^{10}$ and subtract the results $F(1) - F(0)$.
We calculate $F(1)$ and $F(0)$ as follows.
$F(1) = 2 + \frac{2}{25} - \frac{1}{80} = \frac{407}{200}$
$F(0) = 0$
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