Integrate powers and functions of ax + b
Integrate the syllabus forms involving a linear inner expression, standard trigonometric functions and exponentials, with the correct inner factor.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Why does ∫sin(5x) dx include a factor 1/5?
What you will learn
- Integrate (ax + b)ⁿ for rational n, including n = −1
- Integrate sin, cos, sec² and exponential functions of ax + b
- Use radians and include the arbitrary constant
Reverse the chain rule for a linear inner expression
For n ≠ −1, ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n + 1)] + C. The factor a in the denominator reverses the derivative of ax + b.
For n = −1, ∫1/(ax + b) dx = (1/a)ln|ax + b| + C, with a ≠ 0 and ax + b ≠ 0 on the interval considered.
Trigonometric and exponential forms also need the inner factor
∫sin(ax + b) dx = −cos(ax + b)/a + C; ∫cos(ax + b) dx = sin(ax + b)/a + C; ∫sec²(ax + b) dx = tan(ax + b)/a + C.
∫e^(ax+b) dx = e^(ax+b)/a + C. Trigonometric inputs in calculus are in radians, and the calculus formulas are not supplied in the 0606 examination.
Pause and explain
What is ∫sec²(2x) dx?
Worked example
Find ∫(3x + 1)² dx.
Show the worked solution
- Let u = 3x + 1; the outside power integrates to u³/3.
- Since du/dx = 3, divide by the inner factor 3 as well.
- Substitute the original expression and add C.
Answer (3x + 1)³/9 + C.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find ∫sin(2x − 1) dx.
Give me a hint
The derivative of the inner angle is 2.
Compare my reasoning
- The antiderivative of sin u is −cos u.
- Divide by 2 to reverse the inner derivative.
- Add C.
−½ cos(2x − 1) + C.
Look for these in your work
- Sine sign reversed
- Inner factor included
- Constant added
Find ∫e^(4x) dx + ∫cos(3x + π/2) dx.
Give me a hint
Integrate each term with its own linear inner factor.
Compare my reasoning
- ∫e^(4x) dx = e^(4x)/4.
- ∫cos(3x + π/2) dx = sin(3x + π/2)/3.
- Combine and include one arbitrary constant.
e^(4x)/4 + sin(3x + π/2)/3 + C.
Look for these in your work
- Exponential factor included
- Cosine antiderivative correct
- One constant retained
Find ∫[2/(2x + 1) + 3sec²(3x)] dx on an interval where the terms are defined.
Give me a hint
Use the logarithm rule for the first term and reverse the tangent derivative for the second.
Compare my reasoning
- ∫2/(2x + 1) dx = ln|2x + 1|.
- ∫3sec²(3x) dx = tan(3x).
- Add the terms and arbitrary constant.
ln|2x + 1| + tan(3x) + C.
Look for these in your work
- Linear denominator factor handled
- Secant-squared factor cancelled correctly
- Domain and constant respected
Common mistakes
- Forgetting the factor a in the denominator
- Using the power rule for n = −1
- Using the wrong sign for the integral of sine
- Omitting C or treating degree inputs as radians
What can you explain now?
What is ∫e^(4x) dx?
Compare with the explanation
e^(4x)/4 + C
The derivative of the exponent is 4, so divide by 4.
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Make it stick
How do you reverse the chain rule for a function of ax + b?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.