Inverse Trig Functions and Standard Integrals The proof is on page 60 if you wish to read it.

Page 61 Exercise 1A Questions 1(a), (b), (e), (f), 2(a), (c), (f), (h). Page 62 Exercise 1B Questions 1(a), (c), (e), (h) TJ Exercise 1

Integration of Rational Functions.

Page 64 Exercise 2 Questions 5(a), (b), (d), (e), 6(a), (c) Page 66 Exercise 3A Questions 1 to 3 and 7 TJ Exercise 2, 3 and 4

Integration By Parts A method based on the product rule. Sometimes we are asked to integrate the product of two functions. To see how to do this, let us examine the product rule for differentiation. Integrating both sides with respect to x Re-arranging gives: The aim is to make the new integral simpler than the first.

Example 1. Let

Example 2.

Page 69 Exercise 4 Questions 1 and 2 TJ Exercise 5

Developing Integration by Parts Sometimes, the process of integrating by parts must be applied more than once in order to solve the integral. Consider

Let Let

2. Evaluate Let Let

3. Evaluate Note that the derivative of neither function is simpler than the original function. Let Let

Note. When we chose to integrate in the first application of the Rule, we are them committed to integrate in the second application.

Page 71 Exercise 5A Questions 1(a) to (g), (m), (n), (q), (s). TJ Exercise 6

Integration by parts involving a “Dummy” Function Functions like ln(x), sin-1 x, cos-1 x, and tan-1 x do not have a standard integral but have a standard derivative. In order to integrate them, we introduce a “dummy” function, namely the number 1. i.e let f ‘(x) = 1.

Example 1. Let

Example 2. Let For Use the substitution i.e.

TJ Exercise 7 and 8

Differential Equations If an equation contains a derivative then it is called a differential equation. A function which satisfies the equation is called a solution to the differential equation. The order of a differential equation is the order of the highest derivative involved. The degree of a differential equation is the degree of the power of the highest derivative involved.

This section deals only with first order, first degree differential equations. Differential equations are solved by integration. When the solution contains the constant of integration it is called a general solution. When we are given some initial conditions which allow us to evaluate this constant the resultant solution is called a particular solution.

Integrating with respect to x. Thus the particular solution is

Integrating with respect to x. Thus the particular solution is

Further Differential Equations We can now obtain the general solution.

using the initial conditions,

Variables separable

using the initial conditions,

Page 77 Exercise 8 Questions 1, 2, 3 and 5 TJ Exercise 9 and 10

Applications of differential equations Newtons law of cooling states that the rate at which an object cools is proportional to the difference between its temperature and that of its surroundings. Let T be the temperature difference at time t. If T = T0 at t = 0, express T in terms of t.

This is a family of circles centre the origin and radius x y

Page 81 Exercise 9A Questions 2 and 4 to 9 TJ Exercise 11 Do the review on page 86