Solve ∫ (from a to b) of (x-x^2) wrt x

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

Evaluate

$$\frac{a^{3}}{3}-\frac{b^{3}}{3}+\frac{b^{2}}{2}-\frac{a^{2}}{2}$$

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.

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

Simplify.

$$\frac{b^{2}}{2}-\frac{b^{3}}{3}-\frac{a^{2}}{2}+\frac{a^{3}}{3}$$

Differentiate w.r.t. a

$$a\left(a-1\right)$$