14.12 Integrating Standard Functions

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What is Integration?

Integration is the reverse process of differentiation. When you differentiate a function, you find its gradient. When you integrate, you work backwards — you find the original function from the gradient.

Because integration is the reverse of differentiation, the best way to understand each integration rule is to start from what you already know about differentiation and work backwards.

Section 1: Integrating (ax + b)ⁿ

The Rule

When you have a linear expression (something of the form ax + b) raised to a power n, use this formula:

$\int (ax + b)^n \, dx = \frac{1}{a(n+1)}(ax + b)^{n+1} + c$

Conditions:

• $n \neq -1$ (we handle n = −1 separately — see below)
• $a \neq 0$

How to read this formula: Raise the power by 1, then divide by the new power AND divide by a (the coefficient of x). Always add + c.

Where does this rule come from?

Think about differentiation. You know that:

$\frac{d}{dx}\left[\frac{1}{a(n+1)}(ax+b)^{n+1}\right] = \frac{1}{a(n+1)} \times a(n+1)(ax+b)^n = (ax+b)^n$

So working backwards, the integral of $(ax+b)^n$ must be $\dfrac{1}{a(n+1)}(ax+b)^{n+1} + c$.

Step-by-Step Example 1: Positive integer power

Find $\int (3x - 8)^5 \, dx$

Here, $a = 3$, $b = -8$, $n = 5$.

Step 1: Raise the power by 1: $5 + 1 = 6$

Step 2: Write $(3x - 8)^6$

Step 3: Divide by the new power (6) and by $a$ (3): divide by $3 \times 6 = 18$

Answer:

$\int (3x - 8)^5 \, dx = \frac{1}{18}(3x - 8)^6 + c$