Introduction
This O-Level A-Maths question tests your understanding of:
- splitting integrals
- applying the power rule
- integrating expressions of the form \(\dfrac{1}{ax+b}\)
It is a fundamental integration question that students must score confidently in Paper 1.
We follow the teacher’s exact board working.
The Question
Evaluate:
\(\int \left(\dfrac{3}{x^2} + \dfrac{4}{3x – 5}\right) dx\)
Step-by-Step Working (Teacher’s Method)
Step 1: Split the integral
\(\int \dfrac{3}{x^2}\, dx + \int \dfrac{4}{3x – 5}\, dx\)
Step 2: Integrate the first term
Rewrite:
\(\dfrac{3}{x^2} = 3x^{-2}\)
Apply the power rule:
\(\int 3x^{-2}\, dx = 3 \cdot \dfrac{x^{-1}}{-1} = -\dfrac{3}{x}\)
Step 3: Integrate the second term
Let:
\(u = 3x – 5 \quad \Rightarrow \quad du = 3dx\)
So:
\(\int \dfrac{4}{3x – 5}\, dx = \dfrac{4}{3} \ln|3x – 5|\)
Step 4: Combine the results
\(-\dfrac{3}{x} + \dfrac{4}{3}\ln|3x – 5| + C\)
✅ Final Answer
\(\int \left(\dfrac{3}{x^2} + \dfrac{4}{3x – 5}\right) dx = -\dfrac{3}{x} + \dfrac{4}{3}\ln|3x – 5| + C\)
Key Concepts
| Concept | Why It Matters |
|---|---|
| Splitting integrals | Makes integration manageable |
| Power rule | Used for \(x^n\) |
| Logarithmic integration | Required for \(\dfrac{1}{ax+b}\) |
| Constant of integration | Must always be included |
Tips for Students
- Split integrals early
- Rewrite using index notation
- Watch negative powers carefully
- Always include + C
For Parents
Integration is a core A-Maths skill that builds:
- algebra confidence
- calculus fluency
- exam speed
Mastering these basics ensures strong performance in O-Level A-Maths Paper 1.