What is integration in maths, and how do you integrate?

Differentiation takes a function and gives its rate of change, like going from distance to speed. Integration goes the other way: start with the rate of change and get back the original amount, like going from speed to distance. The answer is called an integral (or an antiderivative).

Because x³ differentiates to 3x², the integral of 3x² is x³. But x³ + 5 and x³ − 2 also differentiate to 3x², because constants vanish. So we write

∫3x² dx = x³ + C

The long S, ∫, means "integral of", dx says that x is the variable, and C is the constant of integration, standing for any number. This is an indefinite integral: a family of functions, all with the same gradient.

The power rule for integration reverses the one for differentiation:

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, for any n except −1

In words: add one to the power, then divide by the new power. x² becomes x³/3, x⁴ becomes x⁵/5. A number in front stays (∫5x² dx = 5x³/3 + C), and a sum is integrated term by term. A constant on its own gains an x: ∫7 dx = 7x + C. For n = −1 the rule would divide by 0, so there is a separate result: ∫(1/x) dx = ln|x| + C.

The second big idea is area. Cut the region under a curve into very thin rectangular strips and add up their areas. As the strips get thinner, the total gets closer to the exact area, and that exact area is the definite integral, written with limits a and b:

∫ from a to b of f(x) dx = F(b) − F(a), where F is any integral of f.

This link between areas and reversing differentiation is called the fundamental theorem of calculus. The C cancels when you subtract, so definite integrals give a number, not a formula. Check it on something you know: the area under y = 2x from x = 0 to x = 3 is a triangle with base 3 and height 6, area 9 (see area). By integration: ∫2x dx = x², and 3² − 0² = 9. ✓

Integration turns up wherever a rate is added up over time or space: distance from speed, water in a tank from its flow rate, work done by a force, or the volume of a shape.

Check your answer by differentiating it: you should get back the function you started with.

What is integration in maths?#

It is the reverse of differentiation, and a way to add up infinitely many tiny pieces. Integrating a rate of change gives the total change, and a definite integral gives the area under a curve. (In everyday English, "integration" also means bringing people or systems together, and in software it means combining separate parts or systems so they work together; that is a different topic.)

What are the basic integration rules?#

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫k dx = kx + C; ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C; ∫cos x dx = sin x + C; ∫sin x dx = −cos x + C. Constant multiples stay, and sums are integrated term by term.

What is the integral of a constant?#

∫k dx = kx + C. For example, ∫5 dx = 5x + C. On a graph, the area under the flat line y = 5 from 0 to x is a rectangle of area 5x.

What is integration by parts?#

A rule for integrating some products: ∫u (dv/dx) dx = uv − ∫v (du/dx) dx. For ∫x eˣ dx, take u = x and dv/dx = eˣ: the answer is x eˣ − eˣ + C. It is the reverse of the product rule.

What is integration by substitution (the reverse chain rule)?#

You replace part of the expression with a new letter to make it simpler. For ∫2x(x² + 1)³ dx, let u = x² + 1, so du = 2x dx and the integral becomes ∫u³ du = u⁴/4 + C = (x² + 1)⁴/4 + C.

What are the integrals of tan x, cot x and ln x?#

∫tan x dx = ln|sec x| + C (the same as −ln|cos x| + C), ∫cot x dx = ln|sin x| + C, and ∫ln x dx = x ln x − x + C, which comes from integration by parts.

What is integration used for in physics?#

The area under a velocity–time graph is the displacement, the area under an acceleration–time graph is the change in velocity, and the area under a force–distance graph is the work done. Each one is an integral.