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\int _{2}^{3}3y-y^{2}-3+y\mathrm{d}y
To find the opposite of 3-y, find the opposite of each term.
\int _{2}^{3}4y-y^{2}-3\mathrm{d}y
Combine 3y and y to get 4y.
\int 4y-y^{2}-3\mathrm{d}y
Evaluate the indefinite integral first.
\int 4y\mathrm{d}y+\int -y^{2}\mathrm{d}y+\int -3\mathrm{d}y
Integrate the sum term by term.
4\int y\mathrm{d}y-\int y^{2}\mathrm{d}y+\int -3\mathrm{d}y
Factor out the constant in each of the terms.
2y^{2}-\int y^{2}\mathrm{d}y+\int -3\mathrm{d}y
Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y\mathrm{d}y with \frac{y^{2}}{2}. Multiply 4 times \frac{y^{2}}{2}.
2y^{2}-\frac{y^{3}}{3}+\int -3\mathrm{d}y
Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}. Multiply -1 times \frac{y^{3}}{3}.
2y^{2}-\frac{y^{3}}{3}-3y
Find the integral of -3 using the table of common integrals rule \int a\mathrm{d}y=ay.
2\times 3^{2}-\frac{3^{3}}{3}-3\times 3-\left(2\times 2^{2}-\frac{2^{3}}{3}-3\times 2\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{2}{3}
Simplify.