Integral of $x^{8}$

Q: What is the integral of x^8?

The integral of x^8 is x^99 + C.

The integral of $x^{8}$ is $\frac{x^{9}}{9} + C$ .

\int x^{8}\,dx = \frac{x^{9}}{9} + C

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Graph of f(x) = x^8 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.11111111.

How to work it out

1
We want an antiderivative of f(x) = $x^{8}$ .
$\int x^{8}\,dx$
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:
$\int x^{8}\,dx = \frac{x^{9}}{9} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\frac{x^{9}}{9}\right] = \frac{81 x^{8}}{81}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} x^{8}\,dx = \Big[\frac{x^{9}}{9}\Big]_{0}^{1} \approx 0.11111111$

LaTeX source of the formulas on this page

\[ \int x^{8}\,dx = \frac{x^{9}}{9} + C \]
\[ \int x^{8}\,dx \]
\[ \frac{d}{dx}\left[\frac{x^{9}}{9}\right] = \frac{81 x^{8}}{81} \]
\[ \int_{0}^{1} x^{8}\,dx = \Big[\frac{x^{9}}{9}\Big]_{0}^{1} \approx 0.11111111 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	0.11111111
from 0 to 0.5	0.00021701389

Questions

What is the integral of

x^{8}

The integral of $x^{8}$ is $\frac{x^{9}}{9} + C$ .

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^{8}

from 0 to 1?

About 0.11111111.

Integral of x8x^{8}x8

How to work it out

Definite integrals of the same function

Questions

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Integral of $x^{8}$