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\int _{0}^{2}3x+2x^{2}\mathrm{d}x
Use the distributive property to multiply x by 3+2x.
\int 3x+2x^{2}\mathrm{d}x
Evaluate the indefinite integral first.
\int 3x\mathrm{d}x+\int 2x^{2}\mathrm{d}x
Integrate the sum term by term.
3\int x\mathrm{d}x+2\int x^{2}\mathrm{d}x
Factor out the constant in each of the terms.
\frac{3x^{2}}{2}+2\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 3 times \frac{x^{2}}{2}.
\frac{3x^{2}}{2}+\frac{2x^{3}}{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 2 times \frac{x^{3}}{3}.
\frac{3}{2}\times 2^{2}+\frac{2}{3}\times 2^{3}-\left(\frac{3}{2}\times 0^{2}+\frac{2}{3}\times 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.
\frac{34}{3}
Simplify.