When a standard function has a linear argument ax + b, integrate it as usual and then divide by a, the coefficient of x: ∫ f(ax + b) dx = (1/a)F(ax + b) + c. So (2x + 1)³ integrates to (2x + 1)⁴/8, e^(3x) to ⅓e^(3x), and 1/(2x + 1) to ½ ln|2x + 1|. Forgetting to divide by a is the classic slip.
Integrals of f(ax + b)
- ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / (a(n + 1)) + c, for n ≠ −1
- ∫ e^(ax + b) dx = (1/a)e^(ax + b) + c
- ∫ sin(ax + b) dx = −(1/a)cos(ax + b) + c
- ∫ cos(ax + b) dx = (1/a)sin(ax + b) + c
- ∫ 1/(ax + b) dx = (1/a)ln|ax + b| + c
Why divide by a?
Differentiating a function of ax + b multiplies by a, by the chain rule. Integration reverses that, so it divides by a. You can always check: differentiate your answer, and the factor of a should cancel the 1/a to give back the original integrand.
This only works when the inside is linear. For something like e^(x²), the chain rule factor would be 2x, not a constant, so a different method is needed.
Powers of a linear bracket
Raise the power by one and divide by the new power, then also divide by the coefficient of x. That is two divisions: by n + 1 and by a. There is no need to expand the bracket, which for a high power would be slow and error-prone.
Exponentials and trig with a linear argument
The exponential keeps its form: e^(ax + b) integrates to (1/a)e^(ax + b). For trig, use the standard results and divide by a: sin(ax + b) gives −(1/a)cos(ax + b), and cos(ax + b) gives (1/a)sin(ax + b). Keep the sign change for sine, and work in radians.
The reciprocal of a linear function
1/(ax + b) integrates to (1/a)ln|ax + b|. Keep the modulus, and remember the 1/a. A constant numerator simply multiplies through, so 6/(3x − 1) integrates to 2 ln|3x − 1| + c.
Integrating several linear terms together
When an integrand contains several functions of different linear expressions, deal with each term separately, dividing each by its own coefficient of x. For example, ∫ (e^(2x) + 1/(x + 3)) dx = ½e^(2x) + ln|x + 3| + c. The second term has a = 1, so nothing changes there.
Definite integrals and exact answers
Apply the 1/a rule, then evaluate between the limits. Answers are usually exact expressions involving e, ln or π. Useful simplifications include e^(2 ln 2) = e^(ln 4) = 4 and ln 1 = 0.
Worked examples
Example 1. Find ∫ (2x + 1)³ dx.
- Raise the power and divide by it: (2x + 1)⁴/4.
- Divide by a = 2.
Answer: (2x + 1)⁴/8 + c.
Marks: M1 for raising the power, A1 for (2x + 1)⁴/8 + c.
Example 2. Find ∫ e^(3x) dx.
- The exponential keeps its form.
- Divide by a = 3.
Answer: ⅓e^(3x) + c.
Marks: A1 for ⅓e^(3x) + c.
Example 3. Find ∫ cos 2x dx.
- cos integrates to sin.
- Divide by a = 2.
Answer: ½ sin 2x + c.
Marks: A1 for ½ sin 2x + c.
Example 4. Find ∫ 1/(2x + 1) dx.
- The reciprocal integrates to ln|2x + 1|.
- Divide by a = 2.
Answer: ½ ln|2x + 1| + c.
Marks: A1 for ½ ln|2x + 1| + c.
Example 5. Find the exact value of ∫ e^(2x) dx from 0 to ln 2.
- Integrate: [½e^(2x)] from 0 to ln 2.
- Substitute: ½(e^(2 ln 2) − e⁰).
- e^(2 ln 2) = 4, so this is ½(4 − 1).
Answer: 3/2.
Marks: M1 for ½e^(2x), A1 for 3/2.
Common mistakes
- Forgetting to divide by a. ∫ e^(3x) dx = ⅓e^(3x) + c, not e^(3x) + c.
- Multiplying by a instead of dividing. Differentiation multiplies by a; integration divides.
- Dividing by only one factor for a power. (ax + b)ⁿ integrates to (ax + b)ⁿ⁺¹ divided by both a and n + 1.
- Using the rule when the inside is not linear. For e^(x²) or sin(x²), the 1/a method does not apply.
- Losing the sign change for sin. sin(ax + b) integrates to −(1/a)cos(ax + b).
How integration of f(ax + b) is examined
Integrating functions of ax + b is Pure 3 content (WMA13) in Edexcel International A-Level, and appears in the second year of Edexcel A-Level (9MA0). Typical tasks:
- integrate a power, exponential, trig function or reciprocal of a linear expression
- find exact definite integrals, often with limits involving ln
- find the exact area under a scaled curve such as y = e^(2x) or y = sin 2x
- use the result inside a longer question, such as a volume or a differential equation
Practise exam-style questions, with mark schemes, among the Pure 3 integration exam questions. For further practice, use the integration worksheet or the integration revision engine.
This extends the standard integrals. When the inside is not linear, use the reverse chain rule instead.
Frequently asked questions
How do you integrate e^(ax + b)?
(1/a)e^(ax + b) + c. Keep the exponential and divide by the coefficient of x.
What is the integral of 1/(ax + b)?
(1/a)ln|ax + b| + c.
Can I use this method for (x² + 1)³?
No. The inside must be linear. For a non-linear inside, look for its derivative as a factor and use the reverse chain rule, or expand.