1. Integration
Chapter 9

2. What is Integration?
Method of constructing a function from its derivative
Inverse of differentiation
Suppose F’(x) = f(x). This is equivalent to
𝑓 𝑥 𝑑𝑥 =𝐹 𝑥 +𝐶
C is added to incorporate the fact that derivative of a constant is 0

3. Common Integration Rules
𝑥 𝑎 𝑑𝑥 = 𝑥 𝑎+1 𝑎+1 +𝐶 𝑎≠1
1 𝑥 𝑑𝑥 = ln |𝑥| +𝐶
1 𝑥+𝑎 𝑑𝑥 = ln |𝑥+𝑎| +𝐶
𝑒 𝑎𝑥 𝑑𝑥 = 𝑒 𝑎𝑥 𝑎 +𝐶 (𝑎≠0)
𝑎 𝑥 𝑑𝑥 = 𝑎 𝑥 ln 𝑥 +𝐶 (𝑎>0, 𝑎≠1)

4. General Integration Rules
𝑎𝑓(𝑥)𝑑𝑥 =𝑎∙ 𝑓 𝑥 𝑑𝑥 (𝑎 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
[𝑓 𝑥 +𝑔 𝑥 ] 𝑑𝑥 = 𝑓 𝑥 𝑑𝑥+ 𝑔 𝑥 𝑑𝑥
Integration by substitution: Calculate 2𝑥 ln (𝑥 2 + 𝑎 2 ) 𝑑𝑥

5. Area and Definite Integral
𝒇 𝒙 𝟏 ∆𝒙≤𝐴 𝑥 1 +∆𝑥 −𝐴( 𝑥 1 )≤𝒇( 𝒙 𝟏 +∆𝒙)∆𝒙

6. Definite Integral Properties
𝑎 𝑏 𝑓 𝑥 𝑑𝑥=− 𝑏 𝑎 𝑓 𝑥 𝑑𝑥
𝑎 𝑎 𝑓 𝑥 𝑑𝑥=0
𝑎 𝑏 𝑓 𝑥 𝑑𝑥= 𝑎 𝑐 𝑓 𝑥 𝑑𝑥 + 𝑐 𝑏 𝑓 𝑥 𝑑𝑥
𝑑 𝑑𝑡 𝑎 𝑡 𝑓 𝑥 𝑑𝑥=𝑓(𝑡)
𝑑 𝑑𝑡 𝑎(𝑡) 𝑏(𝑡) 𝑓 𝑥 𝑑𝑥=𝑓 𝑏 𝑡 𝑏 ′ 𝑡 −𝑓 𝑎 𝑡 𝑎′(𝑡)

7. Integration by Parts 𝑓 𝑥 𝑔 ′ 𝑥 𝑑𝑥 =𝑓 𝑥 𝑔 𝑥 − 𝑓 ′ 𝑥 𝑔 𝑥 𝑑𝑥
𝑓 𝑥 𝑔 ′ 𝑥 𝑑𝑥 =𝑓 𝑥 𝑔 𝑥 − 𝑓 ′ 𝑥 𝑔 𝑥 𝑑𝑥
School formula
𝑢 𝑥 𝑣 𝑥 𝑑𝑥 =𝑢 𝑥 𝑣 𝑥 𝑑𝑥 − 𝑢 ′ 𝑥 𝑣 𝑥 𝑑𝑥 𝑑𝑥

8. Infinite intervals for integration
𝑎 ∞ 𝑓 𝑥 𝑑𝑥= lim 𝑏→∞ 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
−∞ 𝑏 𝑓 𝑥 𝑑𝑥= lim 𝑎→−∞ 𝑎 𝑏 𝑓 𝑥 𝑑𝑥

9. Differential Equations
𝑥 𝑡 =𝑓 𝑡
𝑑𝑥 𝑑𝑡 =𝑓 𝑡
𝑑𝑥=𝑓 𝑡 𝑑𝑡
𝑑𝑥 = 𝑓 𝑡 𝑑𝑡

10. Special types of differentiable function
𝑥 𝑡 =𝑟𝑥(𝑡) => 𝑥 𝑡 = 𝑥 0 𝑒 𝑟𝑡
𝑥 𝑡 =𝑎(𝐾 −𝑥 𝑡 ) => 𝑥 𝑡 =𝐾−(𝐾− 𝑥 0 ) 𝑒 −𝑎𝑡
𝑥 𝑡 =𝑟𝑥(𝑡) 1 − 𝑥 𝑡 𝐾 => 𝑥 𝑡 = 𝐾 1+ 𝐾− 𝑥 0 𝑥 0 𝑒 −𝑟𝑡

11. Problem
Let N(t) denote the number of people in a country whose homes have broadband internet.
Suppose that the rate at which new people get access is proportional to the number of people who still have no access.
If the population size is P , the differential equation for N(t) is then 𝑁 (𝑡)= 𝑘(𝑃 − 𝑁(𝑡)) where k is a positive constant.
Find the solution of this equation if N(0) = 0. Then find the limit of N(t) as t → ∞.

12. Problem
A country’s annual natural rate of population growth (births minus deaths) is 2%.
In addition there is a net immigration of persons per year.
Write down a differential equation for the function N(t) which denotes the number of persons in the country at time t (year).
Suppose that the population at time t = 0 is Find N(t).

13. First-order Linear differential equations
𝑥 𝑡 +𝑎 𝑡 𝑥 𝑡 =𝑏 𝑡
a(t) and b(t) are not constants. They are functions of t!
If the equation looks like: 𝑥 𝑡 +𝑎𝑥 𝑡 =𝑏 𝑡 , solution for this differential equation is:
1 𝑒 𝑎𝑡 𝑑 𝑒 𝑎𝑡 𝑥 𝑑𝑡 =𝑏(𝑡)
⇒ 𝑒 𝑎𝑡 𝑥= 𝑏(𝑡) 𝑒 𝑎𝑡 𝑑𝑡
⇒ 𝑥 𝑡 =𝐶 𝑒 −𝑎𝑡 + 𝑒 −𝑎𝑡 𝑒 −𝑎𝑡 𝑏 𝑡 𝑑𝑡
Solve 𝑥 +3𝑥=𝑡 𝑒 𝑡 2 −3𝑡