1. Week 1 Review of the basics 1. Differentiation 2. Integration
3. Sketching of graphs of functions

2. 1. Differentiation
The formulae you are supposed to remember at all times:

3. Example 1:
Find the following derivatives:

4. 2. Integration
The formulae you are supposed to remember at all times:
How to make a substitution x = u(t):

5. Integration by parts:
or, which is the same thing,

6. Example 2:
Find the following integrals:
Example 3:
Calculate

7. Useful formulae:

8. A quick review of partial fractions:
Consider
where P1 and P2 are polynomials in x, and the degree of P1 is strictly smaller than that of P2. Assume also that P2 is factorised.
Then...

9. (unrepeated linear factor)
(unrepeated irreducible quadratic factor)
(repeated irreducible quadratic factor)

10. Example 4: two linear factors
Find

11. Example 5: a repeated linear factor
Find

12. Example 6: an irreducible quadratic
Find

13. Example 7: an improper fraction
Find
Solution:
Observe that the highest degree of x in the numerator is less than that of the denominator. Such fractions are called improper.
To solve the problem, rearrange the integrand using long division and use partial fractions on the remainder.
quotient
denominator
numerator
remainder

14. Now,
remainder
quotient