Integration: powers of x and the reverse of differentiation
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 1 worked example, 5 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- Undoing the speedometer
- Integration reverses differentiation
- The power rule
Read the transcript of the teaching part
Last time
Before we start: last time, in Connected rates of change and small increments, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
Undoing the speedometer
A car's speedometer tells you how fast the distance is changing. Now suppose you only have the speedometer record. Can you recover the distance? That is integration: the reverse of differentiation. You are given the gradient, and you rebuild the curve.
Integration reverses differentiation
If differentiating F gives f, then integrating f gives F plus a constant. The constant c is there because every constant disappears when you differentiate. Always write plus c on an indefinite integral. You can check any integral by differentiating your answer. It should return exactly what you started with, so you always know whether you are right.
The power rule
To integrate x to the n, add one to the power, and divide by the new power. This works for any n except minus one. Integrate x cubed: x to the fourth over four. For minus one, 1 over x integrates to ln x. We meet that next lesson.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Integration as the reverse of differentiation.
The whole chapter: Differentiation, Integration.
Chapter 14: Differentiation and integration
- Differentiation: standard functions, chain, product and quotient rules 5 min
- Tangents and normals 5 min
- Stationary points and the derivative tests 5 min
- Practical maxima and minima 5 min
- Connected rates of change and small increments 5 min
- Integration: powers of x and the reverse of differentiation 5 min
- Integrating (ax + b)ⁿ, exponentials and trigonometric functions 5 min
- Definite integrals and plane areas 5 min
- Kinematics: displacement, velocity, acceleration and graphs 5 min
Also for O Level Additional Maths 4037
Which topics come up most in O Level Additional Maths 4037 past papers, counted from every paper since 2004.