Integration by Parts
∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢
Use LIATE to choose u (log, inverse trig, algebraic, trig, exponential)
L’Hopital’s Rule
Determinate Forms
Indeterminate Forms
0
0
(± ∞) + (± ∞)
0
∞
, 0± ∞ = 0
,
±∞
±∞
, ∞ − ∞, 0 − ∞
1∞, ∞0, 00 : use ln of both sides
Improper Integrals Type 1: (− ∞ to a) or (a to ∞)
1. Find the indefinite integral
2. Calculate the limit as t approaches ∞
a. Change the top bound to t
b. Plug in bounds
c. Calculate limit
If bounds are from − ∞ to ∞, separate it into two integrals
Improper Integrals Type 2: Infinite/Discontinuous Integrand
𝑏
𝑡
1. If lim 𝑓(𝑥) = ∞ then ∫ 𝑓(𝑥) 𝑑𝑥 = lim ∫ 𝑓(𝑥) 𝑑𝑥
−
−
𝑥→𝑏
𝑡→𝑏 𝑎
𝑎
𝑏
𝑏
2. If lim 𝑓(𝑥) = ∞ then ∫ 𝑓(𝑥) 𝑑𝑥 = lim ∫ 𝑓(𝑥) 𝑑𝑥
+
+
𝑥→𝑎
𝑎
𝑡
𝑡→𝑎
3. If f is continuous except at c where a < c < b and lim 𝑓(𝑥) = ∞ then
𝑥→𝑐
𝑏
𝑐
𝑏
∫ 𝑓(𝑥) 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 + ∫ 𝑓(𝑥) 𝑑𝑥
𝑎
𝑎
𝑐
Summary: If a or b is an asymptote, calculate the limit from the right or left.
If there’s an asymptote inside the bounds, separate it into two integrals.
Squeeze Theorem
If lim 𝑎𝑛 = 𝐿 = lim 𝑏𝑛 and 𝑎𝑛 ≤ 𝑐𝑛 ≤ 𝑏𝑛 then lim 𝑐𝑛 = 𝐿
𝑛→∞
𝑛→∞
𝑛→∞
Logistic Growth
𝑦=
𝑦 =population
𝑑𝑦
𝑑𝑥
𝐿
−𝑘𝑡
1+𝐶𝑒
𝑡 =time
𝐿 =carrying capacity
= 𝑘𝑦(1 −
𝑦
𝐿
𝑘 =rate
) = 𝑘𝑦(𝐿 − 𝑦)
k is not the same value for logistic equations and logistic differential equations
At half the carrying capacity, population is growing the fastest (inflection point)
Parametrics
𝑑𝑦
𝑑𝑥
=
2
𝑑 𝑦
2
𝑑𝑥
=
𝑑𝑦/𝑑𝑡
𝑑𝑥/𝑑𝑡
𝑑
𝑑𝑡
(𝑑𝑦/𝑑𝑥)
𝑑𝑥/𝑑𝑡
𝑏
Area under a parametric curve: ∫ 𝑦 𝑥'dt
𝑎
𝑏
Arc length (parametric): ∫
𝑎
𝑥(𝑡) =position
𝑑𝑥 2
( 𝑑𝑡 ) + (
Vectors
𝑥'(𝑡) =velocity
Speed = Length of velocity vector =
𝑑𝑦 2
) dx
𝑑𝑡
𝑥''(𝑡) =acceleration
𝑑𝑥 2
( 𝑑𝑡 ) + (
Total distance traveled = Arc length
𝑑𝑦 2
)
𝑑𝑡
𝑏
Position = Initial position (x or y) + ∫(𝑑𝑥 𝑜𝑟 𝑑𝑦) 𝑑𝑡
𝑎
Polar
𝑥 = 𝑟𝑐𝑜𝑠θ y=rsinθ
𝑑𝑦
𝑑𝑥
=
Area under a polar curve:
1
2
𝑟𝑐𝑜𝑠θ +
𝑑𝑟
𝑑θ
𝑑𝑟
𝑑θ
𝑠𝑖𝑛θ
𝑐𝑜𝑠θ − 𝑟𝑠𝑖𝑛θ
𝑏
2
∫ 𝑟 𝑑θ (bounds are in terms of θ)
𝑎
𝑏
Arc length (polar): ∫
𝑎
𝑑𝑟 2
2
𝑟 + ( 𝑑θ ) 𝑑θ
Finding Area
Find the area on [a, b] of f(x)
𝑛
lim
∑ 𝑓(𝑥𝑖)∆𝑥
𝑛 → ∞ 𝑖=1
∆𝑥 =
𝑏−𝑎
𝑛
𝑥𝑖 = 𝑎 + 𝑖∆𝑥
3
Example: Find the area under 𝑦 = 𝑥 on [2, 4]
∆𝑥 =
𝑛
lim ∑ 𝑓(2 +
𝑛 → ∞ 𝑖=1
4−2
𝑛
2𝑖
𝑛
=
)·
2
𝑛
2
𝑛
𝑥𝑖 = 2 +
2𝑖
𝑛
𝑛
= lim ∑ (2 +
𝑛 → ∞ 𝑖=1
2𝑖 3
)
𝑛
·
2
𝑛