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Summary Notes for Integration and its Application – O Level Add Math

Summary of integration and its application for O level additional Mathematics

Here, you’ll find the summary notes for integration and application of integration written based on what’s tested in the O Level Add Maths syllabus. In my previous articles, I talked about what you need to know about integration and its application for O Levels. You can read about what’s tested for integration and its application in detail here

Summary of integration and its application for O level additional Mathematics

Integration formulae tested in O Level Additional Mathematics

  • integration of constant ; ∫ a dx = ax + c
  • integration of single algebraic term where the power, n is not -1: ∫ axn dx = (axn+1)/(n+1) + c 
  • integration of (ax+b)n where the power of n is not -1: ∫ (ax+b)n dx = (ax+b)n+1/[a(n+1)] + c
  • integration of trigonometric functions:
    • ∫ sin(ax+b) dx = – [cos(ax+b)]/a + c
    • ∫ cos(ax+b) dx =  [sin(ax+b)]/a + c
    • ∫ sec2(ax+b) dx =  [tan(ax+b)]/a + c
  • integration of exponential functions:
    •     ∫ eax+b dx =(eax+b)/a + c
  • integration of algebraic terms with power of -1:
    • ∫ 1/x dx = lnx + c
    • ∫ 1/(ax+b) dx = (1/a)ln(ax+b) + c

Definite integral vs indefinite integral

  • Definite integral i.e. ∫baf(x) dx
    • You’ll get a constant after integrating.
    • baf(x) dx =F(b) – F(a)
  • Indefinite integral: 
    • You’ll get an expression with an arbitrary constant (usually denoted as c) after integrating.
    • baf(x) dx = F(x) + c

Integration as the reverse of differentiation

You also need to know how to apply the knowledge of integration as the reverse of differentiation. Meaning, if you differentiate f(x) and get f'(x), then when you integrate f'(x), you will get f(x) +c.
d/dx[f(x)] = f ‘(x)    <—->      ∫f ‘(x) dx = f(x) + c


Finding the area under the curve by integration

The area bounded by the curve y= f(x), the x- axis, x = a and x =  b based on the diagram below is given by ∫baf(x) dx.

Finding area under the curve and x- axis

The area bounded by the curve x= f(y), the y- axis, y = a and y =  b based on the diagram below is given by ∫baf(y) dy.

area between the curve and y- axis

Finding the area between two curves

The area bounded by the curve y = f(x) and y = g(x) based on the diagram below is given by  ∫b[g(x) – f(x)]dx.

finding area between 2 curves

The area bounded by the curve x = f(y) and x = g(y) based on the diagram below is given by  ∫b[g(y) – f(y)]dy.

finding area between two curves (with respect to the y- axis)

Mathematics is all about applying and being able to do questions. Apart from knowing these notes and formulae, you must also be familiar with how to apply them to the questions. The nice thing about Additional Mathematics is the questions are quite standard.
In my course on integration and its application, I go into detail not only the concepts but how to apply them. We also go through many questions together step-by-step, so that students finish the course equipped with the skills needed to tackle their tests and exams. You can purchase the course instantly on Udemy, and watch and learn instantly. Get our course on integration here

You can find a list of all our Additional Mathematics courses here.

Here’s a visual summary of integration and its application:

Summary of integration for O Level Additional Mathematics
Summary of finding area by integration for o level additional mathematics

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