Solve ∫ (from 0 to 5) of (x-x^2) wrt x

$$\int _ { 0 } ^ { 5 } ( x - x ^ { 2 } ) d x$$

Evaluate

$$-\frac{175}{6}\approx -29.166666667$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

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

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.

$$\frac{5^{2}}{2}-\frac{5^{3}}{3}-\left(\frac{0^{2}}{2}-\frac{0^{3}}{3}\right)$$

Simplify.

$$-\frac{175}{6}$$