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\int 2x-3x^{2}\mathrm{d}x
Evaluate the indefinite integral first.
\int 2x\mathrm{d}x+\int -3x^{2}\mathrm{d}x
Integrate the sum term by term.
2\int x\mathrm{d}x-3\int x^{2}\mathrm{d}x
Factor out the constant in each of the terms.
x^{2}-3\int x^{2}\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 2 times \frac{x^{2}}{2}.
x^{2}-x^{3}
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 -3 times \frac{x^{3}}{3}.
1^{2}-1^{3}-\left(0^{2}-0^{3}\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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