What is the integral of x³?

x⁴/4 + c. Add one to the power, then divide by the new power.

What is the integral of 1/x?

ln x + c. The one power that does not follow the usual rule.

Why is "+ c" added to an indefinite integral?

because any constant differentiates to zero. The original function may have had a constant, and it cannot be recovered without more information.

What is the integral of 1/x²?

−1/x + c. x⁻² becomes x⁻¹ ÷ (−1).

Integration: the reverse of differentiation · Cambridge IGCSE Additional Mathematics

In plain words

Integration undoes differentiation. Given the gradient of a curve, it finds the curve. One thing is lost on the way and can only be found again from extra information: any constant, because every constant differentiates to nothing.

7 things to know

Integration is the reverse of differentiation.
To integrate a power of x: add one to the power, then divide by the new power. xⁿ becomes xⁿ⁺¹ ÷ (n + 1).
The exception is 1/x, the power −1. It integrates to ln x.
An indefinite integral must end with "+ c", the constant of integration.
Integrate a sum one term at a time. A constant multiplier stays in front.
Rewrite roots and fractions as powers first: √x is x^(1/2), and 1/x² is x⁻².
To find c, use a point the curve is known to pass through.

Worked example

Find the integral of 6x² − 4x + 3 with respect to x.

6x² becomes 6x³/3 = 2x³.
−4x becomes −4x²/2 = −2x². 3 becomes 3x.
The answer is 2x³ − 2x² + 3x + c.

Worked example

A curve has dy/dx = 3x² − 2 and passes through (1, 4). Find its equation.

Integrate: y = x³ − 2x + c.
Use the point: 4 = 1 − 2 + c, so c = 5.
y = x³ − 2x + 5.

Tips and tricks

Leaving out "+ c" loses a mark every time.
Check an integral by differentiating it. You should get back what you started with.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Integration: the reverse of differentiation: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the integral of x³?
- 3x²
- x⁴/4 + c (the answer)
- x⁴ + c
- 4x⁴ + c
Add one to the power, then divide by the new power.
What is the integral of 1/x?
- −1/x² + c
- ln x + c (the answer)
- x⁰ + c
- 1 + c
The one power that does not follow the usual rule.
What is the integral of 6x?
- 6 + c
- 3x² + c (the answer)
- 6x² + c
- x² + c
6x²/2.
Why is "+ c" added to an indefinite integral?
- to make the answer bigger
- because any constant differentiates to zero (the answer)
- because the curve passes through the origin
- for neatness
The original function may have had a constant, and it cannot be recovered without more information.
What is the integral of 1/x²?
- ln x² + c
- 1/x + c
- −1/x + c (the answer)
- −2/x³ + c
x⁻² becomes x⁻¹ ÷ (−1).

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 7 marks. Write your answers on paper, then check them.