At the end of this section students should be able to:

1. express a rational function (proper and improper) in partial fractions in the cases where the denominators are:
(a) distinct linear factors,
(b) repeated linear factors,
(c) quadratic factors.
(d) Repeated quadratic factors,
(e) Combinations of (a) and (d) above (repeated factors will not exceed power 2);
2. express an improper rational function as a sum of a polynomial and partial fractions;
3. integrate rational functions in Specific Objectives 1 and 2 above;
4. integrate trigonometric functions using appropriate trigonometric identities;
5. integrate exponential functions and logarithmic functions;
6. find integrals of the form \(\displaystyle \int {f'(x)\over f(x)}\:dx\);
7. use substitutions to integrate functions (the substitution will be given in all but the most simple cases);
8. use integration by parts for combinations of functions;
9. integrate inverse trigonometric functions;
10. derive and use reduction formulae to obtain integrals;
11. use the trapezium rule as an approximation method for evaluating the area under the graph of the function.