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\int \sqrt[3]{x}\mathrm{d}x
Evaluate the indefinite integral first.
\frac{3x^{\frac{4}{3}}}{4}
Rewrite \sqrt[3]{x} as x^{\frac{1}{3}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{3}}\mathrm{d}x with \frac{x^{\frac{4}{3}}}{\frac{4}{3}}. Simplify.
\frac{3}{4}\times 3^{\frac{4}{3}}-\frac{3}{4}\left(-3\right)^{\frac{4}{3}}
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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