5.2 Integration by Substitution

Review Chain Rule

    Find the following Derivatives using the Chain Rule

Chain Rule

Substitution  -  as "the Reverse of the Chain Rule"

The challenge is recognizing g(x) and g '(x)

 #2 Fill in the table by specifying the substitution you could choose to find each of the four given integrals

Integral	Substitution u
a.
b.
c.
d.

Method of Substitution   -   Let u = g(x)  and then find du = g '(x) and substitute

General Indefinite formulas (… with u's!)

 In the text - page 386 - there are "STEPS" for substitution

 Frequently Used Derivatives and Integrals

Applied Problems

#42 Solve the given initial value problem for y = f(x) where y =2 when x=1

#48 The velocity and initial position of an object moving along the x axis is .

a) Find the position x(t) at time t.

b) Find the position of the object at time t = 4.

c) Find the time when the object is at x = 3.

The resale value of a certain industrial machine decreases at a rate that depends on its age. When the machine is t years old, the rate at which its value is changing is dollars per year.

a. Express the value of the machine in terms of its age and initial value.

b. If the machine was originally worth $5,200, how much will it be worth when it is 10 years old?

#58 It is estimated that x years from now, the value V(x) of an acre of farmland will be increasing at the rate of dollars per year. The land is currently worth $500 per acre.

a) Find V(t)

b) How much will the land be worth in 10 years?

c) Use the calculator to determine how long it will take for the land to be worth $1,000 per acre.

Assignment:   5.2; pg 394; 1-29 odd, 35-47 odd, 55, 57, 61