A Level Mathematics: Integration Explained Simply

Quick answer

Integration in A Level Mathematics is about finding a function's original equation from its derivative. It sounds complex, but once you see the patterns, it becomes clearer. Many students freeze during exams, but understanding the basic steps can help you stay calm and score those crucial marks.

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What you need to know

Integration is the reverse process of differentiation. If differentiation is about finding the rate of change, integration helps us find the original quantity before the change. Think of it as piecing back the puzzle from the scattered pieces.

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Understanding Integration

Basic Rules to Remember

1. Integration of a Constant:

   • If you integrate a constant 𝑐, you get cx + 𝐶, where 𝐶 is the constant of integration.
   • Why: Integration "undoes" differentiation, so you add back any lost constants.

2. Power Rule for Integration:

   • When you integrate $𝑥^𝑛$, you get $\frac{𝑥^{𝑛+1}}{𝑛+1} + 𝐶$ (for $𝑛 \neq -1$).
   • Why: This rule helps you find the original polynomial function from its derivative.

Common Integration Formulas

• $\int 𝑥^𝑛 \, dx = \frac{𝑥^{𝑛+1}}{𝑛+1} + 𝐶$ (for $𝑛 \neq -1$)
• $\int 𝑒^𝑥 \, dx = 𝑒^𝑥 + 𝐶$
• $\int \frac{1}{𝑥} \, dx = \ln|𝑥| + 𝐶$

Quick check

Can you integrate these?

1. $\int 3𝑥^2 \, dx$
2. $\int 5 \, dx$
3. $\int 𝑥^{-3} \, dx$

(Answers: 1. $𝑥^3 + 𝐶$, 2. 5𝑥 + 𝐶, 3. $-\frac{1}{2𝑥^2} + 𝐶$)

Common mistakes students make

1. Forgetting the Constant of Integration (𝐶):

   • Always include + 𝐶 at the end of your integration. Missing this can cost you marks.

2. Rushing Algebra Steps:

   • Students often lose marks by not simplifying properly after integration. Slow down, and check your algebra.

3. Misapplying the Power Rule:

   • Remember, the power rule only works for $𝑛 \neq -1$. For 𝑛 = -1, use $\int \frac{1}{𝑥} \, dx = \ln|𝑥| + 𝐶$ instead.

Exam tip

When you see an integration question, immediately think of the power rule or other basic formulas. Recognizing the right formula quickly can save you time and stress. Also, keep your working neat — examiners need to understand your steps to award marks.

Worked examples

Question

Integrate the following: $\int (4𝑥^3 - 3𝑥 + 2) \, dx$

Solution

Step 1: Break the integral into separate terms: $\int 4𝑥^3 \, dx - \int 3𝑥 \, dx + \int 2 \, dx$

Why: It's easier to integrate each term separately.

Step 2: Integrate each term using the power rule:

• $\int 4𝑥^3 \, dx = \frac{4𝑥^4}{4} = 𝑥^4$
• $\int 3𝑥 \, dx = \frac{3𝑥^2}{2}$
• $\int 2 \, dx = 2𝑥$

Why: The power rule helps find the original function from the derivative.

Step 3: Combine the results and add the constant of integration:

• $𝑥^4 - \frac{3𝑥^2}{2} + 2𝑥 + 𝐶$

Why: Combining the terms gives the final integrated function. Don't forget + 𝐶.

Question

Evaluate $\int (5𝑒^𝑥 + \frac{1}{𝑥}) \, dx$

Solution

Step 1: Split the integral: $\int 5𝑒^𝑥 \, dx + \int \frac{1}{𝑥} \, dx$

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Why: Separating terms simplifies integration.

Step 2: Integrate each term:

• $\int 5𝑒^𝑥 \, dx = 5𝑒^𝑥$
• $\int \frac{1}{𝑥} \, dx = \ln|𝑥|$

Why: Use the formula for exponential and logarithmic functions.

Step 3: Combine the results with the constant:

• $5𝑒^𝑥 + \ln|𝑥| + 𝐶$

Why: This step completes the integration by adding + 𝐶.

Quick summary

• Integration reverses differentiation.
• Always include + 𝐶 for indefinite integrals.
• Use the power rule: $\int 𝑥^𝑛 \, dx = \frac{𝑥^{𝑛+1}}{𝑛+1} + 𝐶$.
• Separate terms to simplify integration.
• Recognize when to use specific formulas (exponential, logarithmic).

FAQ

Q: What is the constant of integration?
A: The constant of integration, 𝐶, accounts for any constant that was differentiated away. It's crucial for indefinite integrals.

Q: How do I know which integration rule to use?
A: Identify the form of the function you're integrating. For polynomials, use the power rule. For $𝑒^𝑥$ or $\frac{1}{𝑥}$, use the respective formulas.

Q: Why do I keep losing marks in integration?
A: Common reasons include forgetting + 𝐶, rushing algebra steps, and misapplying rules. Practice slowing down and checking each step.

Q: Is integration always this complicated?
A: It might seem so at first, but with practice, the patterns become clearer. Start with simple functions and build up.

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](/learn/jc-h 2-math-integration)

[Practise unlimited questions on Tutorly.sg/app](https://www.tutorly.sg/app)

Related Topics You Should Learn Next

• [A Level H 2 Math Integration Questions Singapore: A Full Tutorial For Tough Problems](/blog/a-level-h 2-math-integration-questions-singapore)
• [JC H 2 Math Integration Guide](/blog/jc-h 2-math-integration-guide)
• [Integration topic hub](https://tutorly.sg/learn/jc-h 2-math-integration)

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