Solve ∫ (from - 1 to 3) of x(1-2x)

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

Evaluate

$$-\frac{44}{3}\approx -14.666666667$$

Solution Steps

Use the distributive property to multiply x by 1-2x.

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

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -2 times \frac{x^{3}}{3}.

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

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.

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

Simplify.

$$-\frac{44}{3}$$