1. Integration 2 and Differential equations

2. What you need to know
How to integrate trig functions, including using trig identities
Using partial fractions for integration
How to use a substitution to integrate more complicated functions
How to use integration by parts
How to separate variables is a first order differential equation to find general solution
How to use an initial condition to find a particular solution of a differential equation

3. The basic results we need are

4. Integration by substitution
We use u to simplify what we have to integrate, but we must change the dx in terms of du as well as the limits.
We differentiate the u = equation and split the derivative

5. Example 1

6. Example 2

7. Example 3

8. Example 4

9. Example 5

10. Example 6

11. Example 7

12. Integration by parts
– as near as we get to a product rule for integration
 We use the product rule for differentiation to get our formula
We choose one part to be u and the other to be
We can use the word LATE to help us choose –
u is the one that comes first (Log, Algebra, Trig, Exponential)

13. Example 8

14. Example 8

15. Example 9

16. Example 10

17. Example 11

18. Differential equations
General solution has +c as part of the answer
Particular solution gives more information from which c can be found
The key first step is to separate the variables – put all the y’s on left with the dy and all the x’s on the right with the dx

19. Example 12 – general solution

20. Example 13 – Particular integral
Find the particular solution of the differential equation which satisfies x = 0, y = 1

21. Forming and solving differential equations Example 14
Newton’s law of cooling states that the rate of cooling of a body is proportional to the excess temperature., which gives My coffee is 95° when it is made and is initially cooling at 5° per minute when the room is 20°. Find an expression for θ at time t. If I can drink the coffee at 65°, how long do I have to wait before I can drink it?

22. Forming and solving differential equations
My coffee is 95° when it is made and is initially cooling at 5° per minute when the room is 20°. Find an expression for θ at time t.

23. Forming and solving differential equations
If I can drink the coffee at 65°, how long do I have to wait before I can drink it?

24. Using partial fractions and integration Example 15

25. Using partial fractions and integration Example 15

26. Summary Integration of trig and exponential functions
integration by substitution
integration by parts.
The secret is deciding which to use!!
Solving differential equations by separating the variables
Use the boundary conditions to evaluate the constant.