Power Rule (Integral)
Description
The Power Rule for Integration is the reverse operation of the Power Rule for Differentiation. It allows you to find the antiderivative of a function like $x^2$, $x^5$, or $\sqrt{x}$ with a simple two-step process: 1. Add one to the exponent. 2. Divide by the new exponent.
You must also add a **Constant of Integration ($+C$)** because the derivative of any constant is zero, meaning we can't know the original constant value just from the derivative.
**Important Constraint:** This rule works for all real numbers $n$ *except* $n = -1$. If $n = -1$ (i.e., $\frac{1}{x}$), the rule would cause division by zero. The integral of $x^{-1}$ is a special case: $\ln|x| + C$.
History & Origins
The discovery of integration formulas predates differentiation. Bonaventura Cavalieri (1635): An Italian mathematician who used his "method of indivisibles" (slicing shapes into infinitely thin lines) to calculate the area under curves $y=x^n$. He successfully derived the rule for integer powers up to $n=9$. John Wallis (1655): He extended Cavalieri's work, using arithmetic patterns to propose that the rule applied to fractional and negative exponents as well, laying the groundwork for Newton. Bernhard Riemann (1854): While the rule was known, Riemann provided the rigorous definition of the integral (Riemann Sums) that formalized why these shortcuts work.
Proof by Differentiation
The Fundamental Theorem of Calculus states that integration and differentiation are inverse operations. We can prove the integral formula by differentiating the result.
We claim that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.
To prove it, let's differentiate the right side: $\frac{d}{dx} \left( \frac{x^{n+1}}{n+1} + C \right)$.
The derivative of the constant $C$ is 0.
For the term $\frac{1}{n+1} x^{n+1}$, the constant factor $\frac{1}{n+1}$ stays.
Apply the Power Rule for Derivatives to $x^{n+1}$: Bring down $(n+1)$ and subtract 1 from the exponent.
$\frac{d}{dx} (x^{n+1}) = (n+1)x^{(n+1)-1} = (n+1)x^n$.
Multiply by the constant factor: $\frac{1}{n+1} \cdot (n+1)x^n$.
The $(n+1)$ terms cancel out, leaving just $x^n$.
Since the derivative of the result is equal to the integrand ($x^n$), the formula is correct.
Variables
| Symbol | Meaning |
|---|---|
n | Power/exponent (Any real number except -1) |
x | Variable |
C | Constant of Integration |
∫ | Integral symbol (summation) |
Examples
Basic Calculation
Problem: Find the integral of x³
Solution:
Fractional Exponents (Roots)
Problem: Find $\int \sqrt{x} \, dx$.
Solution: $\frac{2}{3}x^{3/2} + C$
- Rewrite radical as exponent: $\sqrt{x} = x^{1/2}$.
- Identify $n = 1/2$.
- Add 1 to exponent: $1/2 + 1 = 3/2$.
- Divide by new exponent: $\frac{x^{3/2}}{3/2}$.
- Simplify division: Dividing by $3/2$ is the same as multiplying by $2/3$.
- Result: $\frac{2}{3}x^{3/2} + C$.
Definite Integral (Area)
Problem: Calculate $\int_0^2 x^2 \, dx$.
Solution: 8/3
- Find antiderivative: $\frac{x^3}{3}$.
- Apply Fundamental Theorem: $F(2) - F(0)$.
- Evaluate at 2: $\frac{2^3}{3} = \frac{8}{3}$.
- Evaluate at 0: $\frac{0^3}{3} = 0$.
- Subtract: $\frac{8}{3} - 0 = \frac{8}{3}$.
Common Mistakes
Forgetting +C
In indefinite integrals, you must add $+C$. If you don't, you represent only one specific curve rather than the whole family of possible solutions.
Using it for n = -1
You cannot calculate $\int x^{-1} dx$ with this rule because you would divide by zero. The correct integral for $\frac{1}{x}$ is $\ln|x| + C$.
Real-World Applications
Physics: From Velocity to Position
In physics, velocity is the derivative of position. To go backwards—to find the position of an object given its velocity function—you integrate. If $v(t) = 3t^2$, then position $x(t) = \int 3t^2 dt = t^3 + C$. The constant $C$ represents the starting position.
Civil Engineering: Beam Deflection
To calculate how much a bridge or beam will bend under a load, engineers start with the load distribution equation and integrate it multiple times (usually 4 times!) using the Power Rule to get the deflection equation.
Frequently Asked Questions
What is C?
C is the "Constant of Integration." It accounts for any constant value that might have been in the original function, because differentiating a constant yields zero.
Why do we add 1 to the exponent?
Because integration is the inverse of differentiation. In derivatives, you subtract 1. To undo that, you must add 1.