The power rule for integration says \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\) for every number \(n\) except \(-1\). In words: add 1 to the exponent, then divide by the new exponent. Example: \(\int x^4\,dx = \frac{x^5}{5} + C\).
The power rule is the first integration rule you learn and the one you will use most. It is simple, but it hides a few traps: negative exponents, square roots, fractions and the one exponent where it fails. This guide teaches it as five levels. Each level adds one new skill, with examples and a quick practice set.
Why the Rule Works
It is the derivative power rule in reverse. The derivative rule takes the exponent down and subtracts 1: \(\frac{d}{dx}x^5 = 5x^4\). To undo that, add 1 to the exponent and divide by the new exponent:
\[\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right) = \frac{(n+1)x^{n}}{n+1} = x^n. \;✓\]
The division by \(n + 1\) is exactly what cancels the number the derivative brings down.
Level 1: Positive Whole-Number Powers
- \(\int x^2\,dx = \frac{x^3}{3} + C\)
- \(\int x^7\,dx = \frac{x^8}{8} + C\)
- \(\int x\,dx = \frac{x^2}{2} + C\) (because \(x = x^1\))
- \(\int 1\,dx = x + C\) (because \(1 = x^0\))
Level 2: Numbers in Front
A coefficient just comes along for the ride (the constant multiple rule). Integrate the power, then multiply.
- \(\int 6x^2\,dx = 6\cdot\frac{x^3}{3} = 2x^3 + C\)
- \(\int 10x^4\,dx = 2x^5 + C\)
- \(\int \frac{x^3}{2}\,dx = \frac{x^4}{8} + C\)
Level 3: Negative Powers
The rule works the same way, but be careful with the signs. Adding 1 to \(-3\) gives \(-2\), not \(-4\).
- \(\int x^{-3}\,dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C\)
- \(\int x^{-2}\,dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C\)
Level 4: Roots and Fraction Powers
Turn every root into a fraction power first: \(\sqrt{x} = x^{1/2}\), \(\sqrt[3]{x} = x^{1/3}\), \(\sqrt{x^3} = x^{3/2}\). Dividing by a fraction is the same as multiplying by its flip.
- \(\int \sqrt{x}\,dx = \int x^{1/2}\,dx = \frac{x^{3/2}}{3/2} = \frac23x^{3/2} + C\)
- \(\int \sqrt[3]{x}\,dx = \frac{x^{4/3}}{4/3} = \frac34x^{4/3} + C\)
- \(\int \frac{1}{\sqrt{x}}\,dx = \int x^{-1/2}\,dx = 2x^{1/2} = 2\sqrt{x} + C\)
Level 5: Rewrite First, Then Integrate
Many problems are power-rule problems in disguise. The rule is: never integrate a fraction or a product of powers directly. Rewrite until every term is a single power of \(x\).
| Original | Rewritten | Answer |
|---|---|---|
| \(\int \frac{4}{x^3}\,dx\) | \(\int 4x^{-3}\,dx\) | \(-\frac{2}{x^2} + C\) |
| \(\int x\sqrt{x}\,dx\) | \(\int x^{3/2}\,dx\) | \(\frac25x^{5/2} + C\) |
| \(\int \frac{x^2 + 1}{\sqrt{x}}\,dx\) | \(\int \left(x^{3/2} + x^{-1/2}\right)dx\) | \(\frac25x^{5/2} + 2x^{1/2} + C\) |
| \(\int (x + 2)^2\,dx\) | \(\int \left(x^2 + 4x + 4\right)dx\) | \(\frac{x^3}{3} + 2x^2 + 4x + C\) |
Putting It Together: Polynomials
With the sum rule, a whole polynomial is just Level 2 repeated:
\[\int \left(5x^4 – 3x^2 + 8x – 2\right)dx = x^5 – x^3 + 4x^2 – 2x + C.\]
Notice there is only one \(+C\) at the end, not one per term.
The Exception: \(n = -1\)
For \(\int x^{-1}\,dx = \int \frac{1}{x}\,dx\), the rule would divide by \(-1 + 1 = 0\). That is impossible, so this case has its own answer: \(\int \frac1x\,dx = \ln|x| + C\). Read why the answer is ln|x|.
What the Power Rule Cannot Do
- Exponentials: \(\int 2^x\,dx\) is not \(\frac{2^{x+1}}{x+1}\). The variable is in the exponent, so it needs the exponential rule.
- Powers of an inside function: \(\int (3x+1)^5\,dx\) needs a quick substitution: \(\frac{(3x+1)^6}{18} + C\). Do not forget to divide by the 3.
- Products of different functions: \(\int x^2\sin x\,dx\) needs integration by parts.
Practice: One Problem per Level
Level 1: \(\int x^9\,dx\)
\(\frac{x^{10}}{10} + C\)
Level 2: \(\int 12x^5\,dx\)
\(2x^6 + C\)
Level 3: \(\int x^{-4}\,dx\)
\(-\frac{1}{3x^3} + C\)
Level 4: \(\int x^{2/3}\,dx\)
\(\frac35x^{5/3} + C\)
Level 5: \(\int \frac{x^3 – 2}{x^2}\,dx\)
\(\int \left(x – 2x^{-2}\right)dx = \frac{x^2}{2} + \frac{2}{x} + C\)
Check any answer with the antiderivative calculator. It names the power rule at every step.
Frequently Asked Questions
What is the power rule for integration?
\(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\) for any real \(n\) except \(-1\). Add one to the exponent and divide by the new exponent.
Does the power rule work for negative exponents?
Yes, as long as the exponent is not \(-1\). For example, \(\int x^{-2}\,dx = -x^{-1} + C = -\frac1x + C\).
How do I use the power rule with square roots?
Rewrite the root as a fraction power first, like \(\sqrt{x} = x^{1/2}\). Then add 1 and divide: \(\int x^{1/2}\,dx = \frac23x^{3/2} + C\).
What is the reverse power rule?
It is another name for the power rule for integration, because it undoes the power rule for derivatives.
Why can’t I use the power rule for 1/x?
Because \(\frac1x = x^{-1}\) and the rule would divide by zero. The antiderivative of \(\frac1x\) is \(\ln|x| + C\) instead.
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