A Level Mathematics: Integration Made Simple

Quick answer

When facing A Level integration questions, it's easy to feel overwhelmed. Many students know the concepts but freeze during exams. This guide will help you recognise patterns, avoid common mistakes, and apply integration techniques confidently.

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What you need to know

Integration is a method in calculus that helps find the area under a curve. It is the reverse process of differentiation. For A Level Math, you need to know different integration techniques like substitution and integration by parts, and when to use each one.

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Basic Integration Techniques

Substitution Method

The substitution method is used when you see a composite function. It's like "undoing" the chain rule from differentiation.

When to use: If the integrand (the function you're integrating) looks like it could be the result of a chain rule, consider substitution.

Step 1: Choose a substitution u = g(x) that simplifies the integrand.

Why: This step helps to transform the integrand into a simpler form, making it easier to integrate.

Step 2: Find du/dx and rearrange to express dx in terms of du.

Why: This allows you to replace dx in the integral, fully converting it in terms of u.

Step 3: Substitute u and du into the integral and solve.

Why: The integral is now in a simpler form that you can integrate directly.

Step 4: Substitute back the original variable x to find the final answer.

Why: Since the question is in terms of x, you need to convert back to match the original variable.

Integration by Parts

Integration by parts is useful when you're dealing with a product of two functions.

When to use: If the integrand is a product of two different types of functions (like polynomial and exponential), use integration by parts.

Step 1: Identify parts of the integrand to be u and dv.

Why: This method relies on the formula: ∫ u dv = uv − ∫ v du.

Step 2: Differentiate u to get du and integrate dv to get v.

Why: You need du and v to apply the integration by parts formula.

Step 3: Substitute into the integration by parts formula and solve.

Why: Applying the formula simplifies the integral, often reducing it to a simpler form.

Step 4: Simplify and solve the remaining integral, if necessary.

Why: Ensure the solution is as complete and simplified as possible.

Quick check

1. Integrate ∫ (2×𝑥 + 1)⁵ dx using substitution.
2. Use integration by parts for ∫ x eˣ dx.
3. Evaluate ∫ sin(x) cos(x) dx.

Common mistakes students make

1. Rushing algebra steps: Many lose marks here. Slow down, write out each step.
2. Wrong substitution choice: Picking the wrong u leads to a complicated integral. Look for what simplifies the most.
3. Forgetting the constant of integration: Always add "+ C" in indefinite integrals. It's part of the solution.

Exam tip

Focus on understanding when to apply each method rather than memorizing formulas. Recognising which technique to use is half the battle. You should immediately think of substitution when you see a chain rule pattern.

Worked examples

Question 1

Integrate ∫ x cos(x²) dx.

Solution

Step 1: Choose u = x². Then, du = 2×𝑥 dx.

Why: The x in the integrand suggests a substitution involving x².

Step 2: Rearrange to find dx = du/(2×𝑥).

Why: You need to express dx in terms of du to substitute.

Step 3: Substitute into the integral: ∫ x cos(u) · du/(2×𝑥).

Why: This simplifies the integral to (1/2) ∫ cos(u) du.

Step 4: Integrate (1/2) sin(u) + C.

Why: Integrating cos(u) gives sin(u), and we include the constant.

Step 5: Substitute back u = x²: (1/2) sin(x²) + C.

Why: Convert back to the original variable for the final answer.

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Question 2

Integrate ∫ x eˣ dx.

Solution

Step 1: Set u = x, dv = eˣ dx.

Why: This setup is ideal for integration by parts.

Step 2: Differentiate u = x to get du = dx and integrate dv = eˣ dx to get v = eˣ.

Why: You need du and v to apply the integration by parts formula.

Step 3: Substitute into the formula: uv − ∫ v du = x eˣ − ∫ eˣ dx.

Why: Applying the formula reduces the integral.

Step 4: Solve: x eˣ − eˣ + C.

Why: The remaining integral ∫ eˣ dx is straightforward to integrate.

Quick summary

• Integration is reverse differentiation.
• Use substitution for composite functions.
• Integration by parts for products of functions.
• Don’t rush; write each step carefully.
• Recognise the pattern for choosing the method.
• Always add "+ C" for indefinite integrals.

FAQ

Q 1: How do I know which integration method to use?
Look for patterns in the integrand. Composite functions suggest substitution, while products of different types of functions suggest integration by parts.

Q 2: What if I make a mistake in my substitution?
If your integral becomes more complicated, rethink your substitution choice. It should simplify the integrand.

Q 3: Why do I keep forgetting "+ C"?
Practice writing it every time. It’s a small detail but important for full marks.

Q 4: Are there shortcuts for integration?
Yes, sometimes recognising patterns or using symmetry can simplify your work. But ensure you practice the full methods too.

Q 5: Can I use a calculator for integration?
Yes, but only to check your work. You need to show all steps in the exam for full credit.

Practise with free question sets

Work through exam-style questions with answers and step-by-step solutions:

• 40+ A Level Integration Questions for 2026/2027 (Singapore MOE Syllabus) — Exam-Style with Full Solutions
• Topic study hub

Practise unlimited questions on Tutorly.sg

Related Topics You Should Learn Next

• A Level H₂ Math Integration Questions Singapore: A Full Tutorial For Tough Problems
• JC H₂ Math Integration Guide
• Topic Cluster Hub: Integration

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