Integral of $x^{5}$

Q: What is the integral of x^5?

The integral of x^5 is x^66 + C.

The integral of $x^{5}$ is $\frac{x^{6}}{6} + C$ .

\int x^{5}\,dx = \frac{x^{6}}{6} + C

Explain it step by step in chat Make a short video of it

Graph of f(x) = x^5 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.16666667.

How to work it out

1
We want an antiderivative of f(x) = $x^{5}$ .
$\int x^{5}\,dx$
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:
$\int x^{5}\,dx = \frac{x^{6}}{6} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\frac{x^{6}}{6}\right] = \frac{36 x^{5}}{36}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} x^{5}\,dx = \Big[\frac{x^{6}}{6}\Big]_{0}^{1} \approx 0.16666667$

LaTeX source of the formulas on this page

\[ \int x^{5}\,dx = \frac{x^{6}}{6} + C \]
\[ \int x^{5}\,dx \]
\[ \frac{d}{dx}\left[\frac{x^{6}}{6}\right] = \frac{36 x^{5}}{36} \]
\[ \int_{0}^{1} x^{5}\,dx = \Big[\frac{x^{6}}{6}\Big]_{0}^{1} \approx 0.16666667 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	0.16666667
from 0 to 0.5	0.0026041667

Questions

What is the integral of

x^{5}

The integral of $x^{5}$ is $\frac{x^{6}}{6} + C$ .

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^{5}

from 0 to 1?

About 0.16666667.

Integral of x5x^{5}x5

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $x^{5}$