Integral of $x^{2}$

Q: What is the integral of x^2?

The integral of x^2 is x^33 + C.

The integral of $x^{2}$ is $\frac{x^{3}}{3} + C$ .

\int x^{2}\,dx = \frac{x^{3}}{3} + C

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Graph of f(x) = x^2 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.33333333.

How to work it out

1
We want an antiderivative of f(x) = $x^{2}$ .
$\int x^{2}\,dx$
2
Method: power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1).
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int x^{2}\,dx = \frac{x^{3}}{3} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\frac{x^{3}}{3}\right] = \frac{9 x^{2}}{9}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} x^{2}\,dx = \Big[\frac{x^{3}}{3}\Big]_{0}^{1} \approx 0.33333333$

LaTeX source of the formulas on this page

\[ \int x^{2}\,dx = \frac{x^{3}}{3} + C \]
\[ \int x^{2}\,dx \]
\[ \frac{d}{dx}\left[\frac{x^{3}}{3}\right] = \frac{9 x^{2}}{9} \]
\[ \int_{0}^{1} x^{2}\,dx = \Big[\frac{x^{3}}{3}\Big]_{0}^{1} \approx 0.33333333 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	0.33333333
from 0 to 0.5	0.041666667

Questions

What is the integral of

x^{2}

The integral of $x^{2}$ is $\frac{x^{3}}{3} + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of

x^{2}

from 0 to 1?

About 0.33333333.

Integral of x2x^{2}x2

How to work it out

Definite integrals of the same function

Questions

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Integral of $x^{2}$