Solve ∫ (from - 3 to 1) of (3-x^2-2x) wrt x

$$\int _ { - 3 } ^ { 1 } ( 3 - x ^ { 2 } - 2 x ) d x$$

Evaluate

$$\frac{32}{3}\approx 10.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3-x^{2}-2x\mathrm{d}x$$

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

$$\int 3\mathrm{d}x-\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x$$

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$3x-\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

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

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}.

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

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.

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

Simplify.

$$\frac{32}{3}$$