Solve ∫ (from 0 to 2) of x(x-1) wrt x

$$\int _ { 0 } ^ { 2 } x ( x - 1 ) d x$$

Evaluate

$$\frac{2}{3}\approx 0.666666667$$

Solution Steps

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

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

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

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

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.

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

Simplify.

$$\frac{2}{3}$$