Solve ∫ (from 0 to a) of (x^2-4x) wrt x

$$\int _ { 0 } ^ { a } ( x ^ { 2 } - 4 x ) d x$$

Evaluate

$$\frac{\left(a-6\right)a^{2}}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}-4x\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -4x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-4\int x\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}-4\int x\mathrm{d}x$$

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 -4 times \frac{x^{2}}{2}.

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

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.

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

Simplify.

$$\frac{\left(-6+a\right)a^{2}}{3}$$

Differentiate w.r.t. a

$$a\left(a-4\right)$$