Transcript AP Calculus AB Chapter 5, Section 2

AP CALCULUS AB
CHAPTER 5, SECTION 2
THE NATURAL LOGARITHMIC FUNCTION:
2013 - 2014
INTEGRATION
LOG RULE FOR INTEGRATION
• Let u be a differentiable function of x.
1
𝑑𝑥 = ln 𝑥 + 𝐶
𝑥
1
𝑑𝑢 = ln 𝑢 + 𝐶
𝑢
ALSO…
• Because 𝑑𝑢 = 𝑢′ 𝑑𝑥, the second formula can also be
written as
𝑢′
𝑑𝑥 = ln 𝑢 + 𝐶
𝑢
USING THE LOG RULE FOR
INTEGRATION
•
2
𝑑𝑥
𝑥
=2
1
𝑑𝑥
𝑥
USING THE LOG RULE WITH A CHANGE
OF VARIABLES
• Find
1
𝑑𝑥
4𝑥−1
FINDING AREA WITH THE LOG RULE
• Find the area of the region bounded by the graph
of
𝑥
𝑦= 2
𝑥 +1
the x-axis, and the line 𝑥 = 3
RECOGNIZING QUOTIENT FORMS OF
THE LOG RULE
3𝑥 2 + 1
𝑑𝑥
3
𝑥 +𝑥
RECOGNIZING QUOTIENT FORMS OF
THE LOG RULE
sec 2 𝑥
𝑑𝑥
tan 𝑥
RECOGNIZING QUOTIENT FORMS OF
THE LOG RULE
𝑥+1
𝑑𝑥
2
𝑥 + 2𝑥
RECOGNIZING QUOTIENT FORMS OF
THE LOG RULE
1
𝑑𝑥
3𝑥 + 2
USING LONG DIVISION BEFORE
INTEGRATING
• Find
𝑥 2 +𝑥+1
𝑑𝑥
𝑥 2 +1
CHANGE OF VARIABLES WITH THE LOG
RULE
• Find
2𝑥
𝑑𝑥
(𝑥+1)2
INTEGRATION
• Integrating is not as straightforward as differentiation.
You have to be able to recognize what is going on
inside the equations. Here are a few guidelines that are
listed in the book.
• Lear a basic list of integration formulas (you have 12: Power
Rule, Log Rule, and 10 trig rules)
• Find an integration formula that resembles all or part of the
integrand to try an figure out what u is. Guess and check here
if needed.
• If you cannot find U substitution, try altering the formula to
make it easier.
U-SUBSTITUTION AND THE LOG RULE
• Solve the differential equation
𝑑𝑦
𝑑𝑥
=
1
𝑥 ln 𝑥
USING A TRIG IDENTITY
• Find
tan 𝑥 𝑑𝑥
INTEGRALS OF THE SIX BASIC
TRIGONOMETRIC FUNCTIONS
sin 𝑢 𝑑𝑢 = − cos 𝑢 + 𝐶
cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶
tan 𝑢 𝑑𝑢 = − ln cos 𝑢 + 𝐶
cot 𝑢 𝑑𝑢 = ln sin 𝑢 + 𝐶
sec 𝑢 𝑑𝑢 = ln sec 𝑢 + tan 𝑢 + 𝐶
csc 𝑢 𝑑𝑢 = − ln csc 𝑢 + cot 𝑢 + 𝐶
INTEGRATING TRIGONOMETRIC
FUNCTIONS
• Evaluate
𝜋/4
0
1 + tan2 𝑥 𝑑𝑥
FINDING AN AVERAGE VALUE
• Find the average value of 𝑓 𝑥 = tan 𝑥 on the
𝜋
interval [0, ]
4
CH 5.2 HOMEWORK
• Pg 338 – 339, #’s: 1 – 41 every other odd, 47, 71, 75
(ignore the direction about Simpson’s rule)