AP Calculus AB and BC

Integral of x^4: Answer, Proof, and Steps

Q: Is the antiderivative of x^4 odd or even?

The antiderivative \frac{x^5}{5} is odd, since x^5 is odd. As a result \int_{-a}^{a} x^4\,dx = 2\int_0^a x^4\,dx, because x^4 itself is even.

The integral of x^4 is x^5/5 + C. By the reverse power rule you add 1 to the exponent to get x^5, then divide by the new exponent 5. Differentiating x^5/5 gives 5x^4/5 = x^4, which confirms the answer.

\int x^4\,dx = \frac{x^5}{5} + C

Applying the reverse power rule

Integrating a power of $x$ reverses the derivative power rule: add $1$ to the exponent and divide by that new exponent.

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1

Here $n = 4$ , so the exponent rises to $5$ and the divisor is $5$ .

\int x^4\,dx = \frac{x^{4+1}}{4+1} + C = \frac{x^5}{5} + C

Divide by the new exponent

After raising $x^4$ to $x^5$ , divide by $5$ , the exponent you just created, not by the original $4$ .

Checking by differentiating

Differentiate the result and confirm it lands back on the integrand.

\frac{d}{dx}\left(\frac{x^5}{5}\right) = \frac{1}{5}\cdot 5x^4 = x^4

The power rule pulls the $5$ down in front, where it cancels the $\frac{1}{5}$ , leaving exactly $x^4$ . The constant $C$ differentiates to $0$ , which is why every antiderivative carries one.

Common mistakes

Dividing by the old exponent. The answer is $\frac{x^5}{5}$ , not $\frac{x^5}{4}$ . The divisor is always the new exponent.
Writing $4x^3$ , which is the derivative of $x^4$ , not its integral. Integration raises the exponent, differentiation lowers it.
Dropping the constant of integration. Every indefinite integral needs $+ C$ .

Every answer on this page is machine checked

An automated test differentiates the antiderivative above and confirms it returns the integrand. A wrong sign or a missing factor fails the build, so it cannot reach you.

Frequently asked questions

What is the integral of x^4 from 0 to 2?

$\int_0^2 x^4\,dx = \left[\frac{x^5}{5}\right]_0^2 = \frac{32}{5} - 0 = \frac{32}{5}$ .

Why do you divide by 5 and not by 4?

Because the reverse power rule divides by the new exponent. Raising $x^4$ to $x^5$ makes the new exponent $5$ , so you divide by $5$ . Differentiating $\frac{x^5}{5}$ brings that $5$ back down and returns $x^4$ .

Is the antiderivative of x^4 odd or even?

The antiderivative $\frac{x^5}{5}$ is odd, since $x^5$ is odd. As a result $\int_{-a}^{a} x^4\,dx = 2\int_0^a x^4\,dx$ , because $x^4$ itself is even.