Integrate sums of powers and reciprocals
Reverse differentiation to integrate sums in powers of x, including x⁻¹, and include an arbitrary constant for indefinite integrals.
Section 14 · Calculus
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 an indefinite integral include + C?
What you will learn
- Interpret integration as the reverse of differentiation
- Integrate powers xⁿ for n ≠ −1
- Integrate x⁻¹ and include the arbitrary constant
An indefinite integral represents a family of functions
If F′(x) = f(x), then ∫f(x) dx = F(x) + C. The constant C appears because every constant differentiates to zero, so differentiating cannot recover the original constant term.
For n ≠ −1, ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C. Increase the power by one and divide by that new power.
The reciprocal power uses a logarithm
When n = −1, the power formula would divide by zero. Instead, ∫1/x dx = ln|x| + C on any interval that does not cross x = 0.
Integrate sums term by term and keep coefficients. The 0606 calculus formulas are not provided in the exam, so the antiderivative rules must be recalled and applied.
Pause and explain
What is the antiderivative of 1/x on x > 0?
Worked example
Find ∫(6x² − 4/x + 3) dx.
Show the worked solution
- Integrate 6x² to get 2x³ and integrate 3 to get 3x.
- Write −4/x as −4x⁻¹; its antiderivative is −4 ln|x|.
- Add one arbitrary constant for the indefinite integral.
Answer 2x³ − 4 ln|x| + 3x + C.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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 ∫(4x³ − 6x + 2) dx.
Give me a hint
Integrate each term separately and add one constant.
Compare my reasoning
- 4x³ integrates to x⁴.
- −6x integrates to −3x²; 2 integrates to 2x.
- Add the arbitrary constant.
x⁴ − 3x² + 2x + C.
Look for these in your work
- Powers increased by one
- Coefficients divided by new powers
- Constant of integration included
Find ∫(3x⁻² + 5/x) dx for x > 0.
Give me a hint
Use the power rule for x⁻² and the logarithm rule for x⁻¹.
Compare my reasoning
- 3x⁻² integrates to −3x⁻¹.
- 5/x integrates to 5 ln x.
- Add C and keep the stated positive domain.
−3/x + 5 ln x + C, for x > 0.
Look for these in your work
- Negative power integrated
- Logarithmic term identified
- Constant and domain retained
A marginal-cost model is C′(x) = 4x + 6/x for x > 0. Find the cost function C(x), allowing an unknown fixed cost.
Give me a hint
Integrate both terms and represent the fixed cost with C.
Compare my reasoning
- ∫4x dx = 2x².
- ∫6/x dx = 6 ln x for x > 0.
- Add the arbitrary constant representing fixed cost.
C(x) = 2x² + 6 ln x + K, for x > 0.
Look for these in your work
- Marginal terms integrated
- Logarithm domain respected
- Fixed constant interpreted
Common mistakes
- Forgetting the arbitrary constant C
- Using the power rule when n = −1
- Dividing by the original power instead of the new power
- Integrating a sum but changing a coefficient or sign
What can you explain now?
What is ∫x⁴ dx?
Compare with the explanation
x⁵/5 + C
Increase the power to 5 and divide by 5.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What are the power-rule antiderivative and its special case for x⁻¹?
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.