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\int \frac{-27x^{2}+9x^{3}}{20}\mathrm{d}x
Evaluate the indefinite integral first.
\int -\frac{27x^{2}}{20}\mathrm{d}x+\int \frac{9x^{3}}{20}\mathrm{d}x
Integrate the sum term by term.
\frac{-27\int x^{2}\mathrm{d}x+9\int x^{3}\mathrm{d}x}{20}
Factor out the constant in each of the terms.
\frac{-9x^{3}+9\int x^{3}\mathrm{d}x}{20}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -1.35 times \frac{x^{3}}{3}.
-\frac{9x^{3}}{20}+\frac{9x^{4}}{80}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 0.45 times \frac{x^{4}}{4}.
-\frac{9}{20}\times 4^{3}+\frac{9}{80}\times 4^{4}-\left(-\frac{9}{20}\times 0^{3}+\frac{9}{80}\times 0^{4}\right)
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
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Simplify.
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