The Power Rule for Antiderivatives

Reversing the Derivative Power Rule

Differentiation turns powers into new powers: for a positive integer $n$, the derivative of $x^n$ is $nx^{n-1}$. Antidifferentiation reverses that change. If we want a function whose derivative is $x^n$, we need to raise the exponent by one and choose a coefficient that cancels the factor produced by differentiation.

This gives a reliable rule for finding antiderivatives of powers. The rule applies when the exponent is not $-1$; the exponent $-1$ is an exception because the rule’s denominator would be zero.

Power rule for antiderivatives: For a real number $n\ne -1$, on the interval $x>0$, $$ \int x^n\,dx=\frac{x^{n+1}}{n+1}+C. $$ Here $C$ is an arbitrary constant. For nonnegative integer $n$, the same formula applies for all real $x$. This tutorial’s worked examples use nonnegative integer powers.

An indefinite integral represents a family of functions. In this rule, $C$ accounts for the fact that adding a constant does not change a derivative. The result is an antiderivative because differentiating it returns the integrand.

Why the Rule Works

The rule follows directly from the derivative power rule. Begin with the expression suggested by increasing the exponent of $x^n$ by one. Differentiating that expression produces a factor of $n+1$, so dividing by $n+1$ beforehand cancels that factor.

\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right) = \frac{1}{n+1}(n+1)x^n = x^n, \qquad n\ne -1.

The condition $n\ne -1$ is needed because the proposed coefficient $1/(n+1)$ is defined only when $n+1\ne 0$. This calculation proves that the displayed expression is an antiderivative. To see why it gives the full family, suppose $F$ is any other antiderivative of $x^n$ on an interval. Let $G(x)=x^{n+1}/(n+1)$. Since $F'(x)=x^n$ and $G'(x)=x^n$, the difference $H=F-G$ satisfies

$$ H'(x)=F'(x)-G'(x)=x^n-x^n=0. $$

For any two points $a$ and $b$ in that interval, the Mean Value Theorem applies to $H$ on the segment between them. It gives a point $c$ between $a$ and $b$ such that

$$ H(b)-H(a)=H'(c)(b-a)=0. $$

Thus $H(b)=H(a)$ for every pair of points in the interval, so $H$ is constant there. Therefore every antiderivative has the form $G(x)+C$. This justifies both the formula and the constant of integration.

A Repeatable Method

For a monomial $a x^n$, where $a$ is a constant and $n\ne -1$, keep the coefficient $a$, raise the exponent by one, and divide by the new exponent. Then add $C$. The coefficient remains because the integral is linear with respect to constant multiples.

\int a x^n\,dx = a\frac{x^{n+1}}{n+1}+C.

A quick check is part of the method: differentiate the proposed answer and confirm that the result is exactly the original integrand. The check catches errors in either the new exponent or its coefficient.

Check the exponent.
Confirm that $n\ne -1$, so the new exponent $n+1$ is not zero.

Increase the exponent by one.
Replace $x^n$ with $x^{n+1}$ in the antiderivative.

Divide by the new exponent.
Multiply by $1/(n+1)$ to undo the factor created by differentiation.

Add $C$ and check.
Include the arbitrary constant, then differentiate the result.

Worked Examples

Worked Example: An Antiderivative of $7x^4$

Find $\int 7x^4\,dx$. The coefficient is $7$, and the exponent is $4$, so the new exponent is $4+1=5$. Divide by $5$:

\int 7x^4\,dx = 7\frac{x^{4+1}}{4+1}+C = \frac{7}{5}x^5+C.

Check by differentiation. The derivative of $C$ is $0$, and the constant multiple rule gives

\frac{d}{dx}\left(\frac{7}{5}x^5+C\right) = \frac{7}{5}\cdot 5x^4+0 = 7x^4.

The derivative matches the original integrand, so the family of antiderivatives is $\frac{7}{5}x^5+C$.

Worked Example: A Negative Coefficient

Find $\int -12x^2\,dx$. Increase the exponent $2$ to $3$, and divide the coefficient $-12$ by $3$:

\int -12x^2\,dx = -12\frac{x^{2+1}}{2+1}+C = -12\frac{x^3}{3}+C = -4x^3+C.

Differentiate to check:

\frac{d}{dx}(-4x^3+C) = -4\cdot 3x^2+0 = -12x^2.

The coefficient is negative in both the integrand and its verified derivative. The negative sign does not change the steps of the power rule.

Worked Example: A Constant Function

Find $\int 9\,dx$. Write the constant function as a multiple of $x^0$, since $x^0=1$:

$$ 9=9x^0. $$

The exponent $0$ increases to $1$, and dividing by $1$ leaves the coefficient unchanged:

\int 9\,dx = \int 9x^0\,dx = 9\frac{x^{0+1}}{0+1}+C = 9x+C.

The check is

\frac{d}{dx}(9x+C)=9+0=9.

Thus $9x+C$ is the complete family of antiderivatives of the constant function $9$.

The Exception at Exponent $-1$

The condition $n\ne -1$ is essential. If $n=-1$, then increasing the exponent by one gives zero, and dividing by the new exponent would mean dividing by zero:

\frac{x^{(-1)+1}}{(-1)+1} = \frac{x^0}{0}.

That expression is undefined, so the power rule cannot be used at exponent $-1$. The antiderivative for that exponent is handled by a different rule. For this topic, the important point is simply to check the exponent before applying the power rule; do not substitute $-1$ into the formula.

Common Mistake: Changing the Exponent but Not the Coefficient

A frequent error is to raise the exponent by one but forget to divide by the new exponent. For instance, an incorrect attempt at $\int x^4\,dx$ might be

\int x^4\,dx \mathrel{\not=} x^5+C.

Differentiating the proposed answer exposes the mistake:

\frac{d}{dx}(x^5+C)=5x^4,

which is five times the required integrand. The correction is to divide by $5$:

\int x^4\,dx=\frac{x^5}{5}+C, \qquad \frac{d}{dx}\left(\frac{x^5}{5}+C\right)=x^4.

Another error is to forget $C$. For example, $\frac{x^5}{5}$ is one antiderivative of $x^4$, but it is not the complete family. Every function $\frac{x^5}{5}+C$ has the same derivative, so an indefinite integral must include the arbitrary constant.

Why the Rule Is Useful

The power rule provides a direct way to reverse differentiation for powers without guessing and adjusting coefficients. Its structure is worth remembering as a pair of linked steps: increase the exponent by one, then divide by that new exponent. The derivative check explains why both steps are needed.

When applying the rule, keep the exponent condition visible, preserve any constant coefficient, and write $+C$. Then differentiate the result. That final check is especially useful when the coefficient is a fraction or a negative number, because it verifies the arithmetic as well as the calculus.

Key takeaway: For $n\ne -1$, the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}+C$. Raising the exponent creates a factor of $n+1$ when differentiated; dividing by that factor makes the derivative equal the original power.

Check Your Understanding

For each question, apply the power rule where it is permitted and use differentiation to check your result.

Find $\int 6x^3\,dx$, and verify the answer by differentiation.
Find $\int -5x^5\,dx$, showing the new exponent and coefficient.
Rewrite the constant function $4$ as a multiple of $x^0$, then find its antiderivatives.
What goes wrong if you substitute $n=-1$ into the power rule?
Explain why differentiating $x^6+C$ shows that it is not an antiderivative of $x^5$.

The Power Rule for Antiderivatives

What You'll Learn

Reversing the Derivative Power Rule

Why the Rule Works

A Repeatable Method

Worked Examples