MATH141: Mathematics 1C Part 1

Overview of Lectures on integration

In this section I will try to summarise the main ideas of each lecture.

Before you start revising integration it is a good idea to revise basic differentiation.

Week 10, lecture 2 (The indefinite integral)

1. Suppose that dy/dx = 1. True or false: the function y is given by y=x?
2. If d/dxF(x) = f(x) then
   ∫ f(x)dx = F(x) +c, where c is a constant of integration.

Week 11, lecture 1 (the definite integral)

1. Let F(x) be the antiderivative of the function f(x). Then
   ∫_a^b f(x)dx = F(b) -F(a). (This is the definite integral of f(x) with lower limit x=a and upper limit x=b. The HTML code doesn't come out quite as I'd like it to).
2. The limits of integration may be functions of another parameter. For instance, we may have a=g(x) and b=h(x). In this case
   ∫_g(x)^h(x) f(t)dt = F(h(x)) -F(g(x)).
3. The following result is useful
   d/dx ∫_a^g(x) f(t)dt = f(g(x))× g'(x) .
   Note, we do not need to integrate the function f(t) to do this calculation. In fact, it may be impossible to integrate the function f(t).

Week 11, lecture 2 (properties of integrals; methods of integration)

Properties of Integrals. The list of properties is unimportant for MATH141, they will be more important in MATH142. Most of these `abstract' looking properties make sense if you think of integrals in terms of areas under a curve.

Methods of Integration.

1. Integrals by Inspection. If you are good at differentiating then you will be able to integrate many functions "by inspection".
2. Simplifying Integrals. Sometimes you need to simplify an integrand before it becomes sufficiently easy to integrate "by inspection".
3. Using integral tables. There's generally been two integrals on the exam paper that use tables. Learn how to use the tables for some easy marks.

Week 12, lecture 1 (algebraic substitution)

1. Integrals of the form ∫ g(f(x))×f'(x)dx (and other types of integrals) may be solved by making a substitution. Here the substitution is u = f(x).
   1. You always have to differentiate the substitution so that you can find dx in terms of du.
   2. By replacing dx by a suitable expression involving du and making the substitution u = f(x). You will obtain an integral ∫ f(u)du that you can (hopefully) integrate.
   3. Evaluate the integral. (Don't forget the constant of integration!). Your answer will be some function of u, say G(u).
   4. Back-substitute so that your final answer is in terms of x. G(u) = G(f(x)) = F(x)
2. Consider the definite integral ∫_a^b g(f(x))×f'(x)dx. We can do this in two ways.
   1. When we change variables from x to u we need to change the limits. When x=a we have u = f(a) and when x=b we have u = f(b). Thus our problem becomes
      ∫_u=f(a)^u=f(b) g(u)du = G(b) - G(a).
   2. Ignore the limits and consider the indefinite integral ∫ g(f(x))×f'(x)dx. Evaluate the integral (by substitution) to obtain the integrand F(x). Then
      ∫_a^b g(f(x))×f'(x)dx = F(b)-F(a)
3. Integrals of the form ∫ f'(x)/f(x)dx are solved by making the substitution u=f(x) to obtain the integral
   ∫ (1/u)du = ln|u| +c = ln|f(x)| +c

Week 12, lecture 2 (revision of integrals)

We went through some practice questions on integration.