Solve ∫ (from 0 to 2) of x(x-2) wrt x

$$\int _ { 0 } ^ { 2 } x ( x - 2 ) d x$$

Evaluate

$$-\frac{4}{3}\approx -1.333333333$$

Solution Steps

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

$$\int _{0}^{2}x^{2}-2x\mathrm{d}x$$

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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}.

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

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.

$$\frac{2^{3}}{3}-2^{2}-\left(\frac{0^{3}}{3}-0^{2}\right)$$

Simplify.

$$-\frac{4}{3}$$