Integration methods

Integration is differentiation run backwards. Given a gradient formula, you rebuild the function it came from, and given two limits you can find the signed area under a curve.

It leads straight into areas, kinematics and any question that starts from a rate. Confirm the exact list of functions you must integrate on the current 0606 syllabus page before you revise, because the lessons here stop at what that page lists.

What should you know before starting?

You need secure differentiation, because every integration answer is checked by differentiating it. Revisit differentiation techniques if the power rule or the chain rule still feels uncertain.

You also need indices. Every term must be written as a power of x before you integrate, so 1/x² = x⁻² and √x = x^1/2. Comfort with tangents, normals and rates helps, since a rate of change is what you are given to integrate in many questions.

An orienting worked example

A curve has gradient dy/dx = 6x² − 4 and passes through (1, 3). Find its equation.

Step 1, integrate each term: raise the power by one and divide by the new power. y = 2x³ − 4x + c.

Step 2, use the point: substitute x = 1, y = 3. 3 = 2 − 4 + c, so c = 5.

Step 3, write the answer: y = 2x³ − 4x + 5.

Step 4, check: differentiate to get 6x² − 4, and test x = 1: 2 − 4 + 5 = 3. Both agree.

Without the point, the answer would be a whole family of curves. The point is what fixes the constant.

In what order should you study the lessons?

1. Recover a function from its derivative and a point: the core idea, and the reason the constant exists.
2. Integrate a power with the correct constant: negative and fractional powers, where most slips happen.
3. Use a substitution-shaped reverse derivative: brackets such as (3x − 2)⁴ and e^2x+1, which need a divide by the inner coefficient.
4. Evaluate a definite integral with correct limits: upper minus lower, and what a negative answer means.
5. Check an antiderivative by differentiation: the habit that makes every earlier answer safe.

Then try the integration methods practice set. The calculus shape and rate explorer lets you see a curve, its integral and the signed area together.

What are the common traps?

• Forgetting + c in an indefinite integral, which costs the method mark for finding the constant later.
• Dividing by the old power instead of the new power.
• Ignoring the inner coefficient in a bracket, so (3x − 2)⁵ is not divided by 15.
• Substituting limits the wrong way round, or dropping the brackets around a negative lower limit.
• Treating a negative definite integral as an error, when it simply means the area lies below the axis.

How should you use the practice set?

Attempt each question on paper first. Write the integrated expression on its own line before you substitute anything.

Afterwards, differentiate two or three of your answers to confirm them. The “If you got these wrong” section points each error type back to one lesson.

If it is method choice rather than the rules that holds you back, our online one-to-one Additional Mathematics tuition can focus on exactly that, and the Additional Mathematics learning guide shows where integration sits in the course.