Solve f ( x ) = ∫ (from 0 to x) of (t^2-t)dt

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

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}.

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

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t\mathrm{d}t with \frac{t^{2}}{2}. Multiply -1 times \frac{t^{2}}{2}.

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

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.

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

Simplify.

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

Differentiate w.r.t. x

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