Integration 2 and Differential equations

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

The basic results we need are

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

Example 1

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Example 7

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)

Example 8

Example 8

Example 9

Example 10

Example 11

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

Example 12 – general solution

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

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?

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.

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?

Using partial fractions and integration Example 15

Using partial fractions and integration Example 15

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.