O Level Additional Mathematics: Integration Worked Examples Step by Step

Quick answer

Integration in O Level Additional Mathematics often looks daunting at first, but with practice, you'll see it's about recognizing patterns and applying the right steps. When you face a question that seems different from what you’ve practiced, remember: the core concepts remain the same. This guide will walk you through four integration problems step by step, helping you understand why each move is made.

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What you need to know

Integration is a mathematical process used to find the area under a curve. In simpler terms, it's like the reverse of finding the slope of a line. You’ll often see this in exam questions where you need to find the area between a curve and the x-axis. The main tool you'll use is the integration formula, which helps you work backward from the derived function.

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Common mistakes students make

1. Rushing through steps: Many students lose marks because they skip steps in their rush to complete the exam. Remember, each step is crucial.

2. Misapplying formulas: Sometimes, students use the wrong formula because they panic. Take a moment to identify the type of question before starting.

3. Forgetting constants: When integrating, don't forget to add the constant of integration, +C. This is where many students lose unnecessary marks.

4. Incorrect substitution: When using substitution, double-check your limits if you're working with definite integrals. This part trips everyone up.

Exam tip

Always allocate time to check your work. After completing an integration question, quickly run through each step again to ensure no careless mistakes. Presentation matters too: write neatly and clearly to avoid misreading your own work under pressure.

Worked examples

Question 1

Integrate $𝑓(𝑥) = 3𝑥^2$.

Solution

Step 1: Identify the formula for integration.
Why: We need to know which formula fits the function we have. Here, $\int 𝑥^𝑛 \, dx = \frac{𝑥^{𝑛+1}}{𝑛+1} + 𝐶$.

Step 2: Apply the formula to the function $3𝑥^2$.
Why: This function is in the form $𝑥^𝑛$ where 𝑛 = 2.

Step 3: Integrate: $\int 3𝑥^2 \, dx = \frac{3𝑥^{2+1}}{2+1} + 𝐶 = 𝑥^3 + 𝐶$.
Why: We integrate each term separately and simplify the expression.

Question 2

Integrate $𝑓(𝑥) = 5𝑥^3 - 2𝑥 + 7$.

Solution

Step 1: Break the function into separate terms: $5𝑥^3$, -2𝑥, and $7$.
Why: Each term is integrated separately using the power rule.

Step 2: Integrate each term:
$\int 5𝑥^3 \, dx = \frac{5𝑥^{3+1}}{3+1} = \frac{5𝑥^4}{4}$,
$\int -2𝑥 \, dx = \frac{-2𝑥^{1+1}}{1+1} = -𝑥^2$,
$\int 7 \, dx = 7𝑥$.

Why: We apply the power rule to each term, simplifying where possible.

Step 3: Combine all the integrated terms:
$\frac{5𝑥^4}{4} - 𝑥^2 + 7𝑥 + 𝐶$.
Why: Summing the integrated terms gives the complete integral.

Question 3

Integrate $𝑓(𝑥) = \frac{1}{𝑥}$.

Solution

Step 1: Recognize this as a special case where $\int \frac{1}{𝑥} \, dx = \ln |𝑥| + 𝐶$.
Why: This is a well-known integral form that doesn’t follow the power rule.

Step 2: Apply the formula: $\int \frac{1}{𝑥} \, dx = \ln |𝑥| + 𝐶$.
Why: The function fits the formula directly, so we use it.

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Question 4

Evaluate the definite integral $\int_{1}^{3} (2𝑥 + 1) \, dx$.

Solution

Step 1: Integrate the function 2𝑥 + 1 without limits.
Why: First, find the indefinite integral before applying limits.

Step 2: Integrate: $\int (2𝑥 + 1) \, dx = 𝑥^2 + 𝑥 + 𝐶$.
Why: Integrate each term separately.

Step 3: Apply the limits:

Related Topics You Should Learn Next

• O Level Additional Mathematics: Integration Made Simple
• O Level Additional Mathematics Integration: Your Guide to Avoiding Common Pitfalls
• [A Level H 2 Math Integration Questions Singapore: A Full Tutorial For Tough Problems](https://tutorly.sg/blog/a-level-h 2-math-integration-questions-singapore)
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