Solve ∫ (from - 1 to 2) of (y^2-2)-y wrt y

$$\int _ { - 1 } ^ { 2 } ( y ^ { 2 } - 2 ) - y d y$$

Evaluate

$$-\frac{9}{2}=-4.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int y^{2}-2-y\mathrm{d}y$$

Integrate the sum term by term.

$$\int y^{2}\mathrm{d}y+\int -2\mathrm{d}y+\int -y\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int y^{2}\mathrm{d}y+\int -2\mathrm{d}y-\int y\mathrm{d}y$$

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}.

$$\frac{y^{3}}{3}+\int -2\mathrm{d}y-\int y\mathrm{d}y$$

Find the integral of -2 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$\frac{y^{3}}{3}-2y-\int y\mathrm{d}y$$

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 -1 times \frac{y^{2}}{2}.

$$\frac{y^{3}}{3}-2y-\frac{y^{2}}{2}$$

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.

$$\frac{2^{3}}{3}-2\times 2-\frac{2^{2}}{2}-\left(\frac{\left(-1\right)^{3}}{3}-2\left(-1\right)-\frac{\left(-1\right)^{2}}{2}\right)$$

Simplify.

$$-\frac{9}{2}$$