Integral of x²: Formula, Derivation and Examples

The integral of x^2 is \dfrac{x^3}{3} + C. Add one to the exponent and divide by the new exponent.

Graph: the area under y = x²

Notice that the antiderivative is flat at x = 0 (where x^2 = 0) and never decreases, because x^2 is never negative.

Derivation with the reverse power rule

1. Start from the reverse power rule: \int x^n\,dx = \frac{x^{n+1}}{n+1} + C for n \ne -1.
2. Set n = 2: the new exponent is 3.
3. Divide by 3: \frac{x^3}{3}.
4. Add C.

A proof from Riemann sums

You can also find the area under x^2 from 0 to b without knowing any antiderivative. Split [0, b] into n strips of width \frac{b}{n} and use right endpoints x_i = \frac{bi}{n}:

As n \to \infty, the fraction \frac{n(n+1)(2n+1)}{6n^3} tends to \frac{2}{6} = \frac{1}{3}, so the area is \frac{b^3}{3}. That matches the antiderivative. This is the same computation Archimedes did for the parabola, and you can practise it on the Riemann sums guide.

Check by differentiating

See also the derivative of x².

Substitution variant: integral of (ax + b)²

With u = ax + b and du = a\,dx:

For example, \int (2x + 1)^2\,dx = \frac{(2x + 1)^3}{6} + C. Check: the derivative is \frac{3(2x+1)^2\cdot 2}{6} = (2x + 1)^2.

The same idea handles an extra factor of x: \int x(x^2 + 1)^2\,dx = \frac{(x^2 + 1)^3}{6} + C, using u = x^2 + 1. More on this on the u-substitution rule page.

Worked examples

Example 1. \int_1^3 x^2\,dx = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}.

Example 2. \int (x^2 + 3x - 2)\,dx = \frac{x^3}{3} + \frac{3x^2}{2} - 2x + C.

Example 3. \int_{-2}^{2} x^2\,dx = \frac{8}{3} - \left(-\frac{8}{3}\right) = \frac{16}{3}. Since x^2 is even, this is twice \int_0^2 x^2\,dx.

Example 4: volume. Spinning y = x around the x-axis on [0, h] gives a cone with volume \pi\int_0^h x^2\,dx = \frac{\pi h^3}{3}, the familiar \frac{1}{3}\pi r^2 h with r = h.

Common mistakes

1. Writing 2x. That is the derivative, not the integral.
2. Forgetting to divide by 3. x^3 has derivative 3x^2, not x^2.
3. Integrating (2x + 1)^2 as \frac{(2x+1)^3}{3}. You must also divide by the inside derivative, 2.
4. Squaring the integral of x. Integrals do not distribute over products.

Practice with the antiderivative problems and definite integral problems.