14.11 Integrating Powers of x

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Main Content

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What is Integration?

Integration is the reverse process of differentiation. If differentiation takes a function and finds its gradient (rate of change), integration works backwards — it takes a gradient and finds the original function.

The symbol for integration is $\int$ (a stretched letter "S", which stands for "sum"). You write it like this:

$\int x^2 \, dx$

• The $\int$ tells you to integrate
• $x^2$ is the function you are integrating
• $dx$ tells you that $x$ is the variable you are integrating with respect to

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The Arbitrary Constant c

Every time you differentiate, the constant disappears. For example:

$y = x^3 + 7 \quad \Rightarrow \quad \frac{dy}{dx} = 3x^2$ $y = x^3 - 4 \quad \Rightarrow \quad \frac{dy}{dx} = 3x^2$

Both give the same derivative $3x^2$, but they had different constants. So when we reverse the process (integrate), we do not know what the original constant was. We write + c to represent this unknown constant.

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Rule 1 — The Power Rule for Integration

This is the main rule for integrating powers of x:

$$\boxed{\int x^n \, dx = \frac{x^{n+1}}{n+1} + c, \quad \text{where } n \neq -1}$$

In plain English:

Why $n \neq -1$? If $n = -1$, then $n + 1 = 0$, and you would be dividing by zero — which is impossible. This special case is handled separately below.

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Step-by-Step: Using the Power Rule

Example 1: Find $\displaystyle\int x^4 \, dx$

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