THE NATURAL LOGARITHMIC FUNCTION: INTEGRATION 2013 - 2014 AP CALCULUS AB CHAPTER 5, SECTION 2
Dec 26, 2015
T H E N A T U R A L L O G A R I T H M I C F U N C T I O N : I N T E G R A T I O N2 0 1 3 - 2 0 1 4
AP CALCULUS ABCHAPTER 5, SECTION 2
LOG RULE FOR INTEGRATION
• Let u be a differentiable function of x.
ALSO…
• Because , the second formula can also be written as
USING THE LOG RULE FOR INTEGRATION
∫ 2𝑥𝑑𝑥=2∫ 1
𝑥𝑑𝑥
USING THE LOG RULE WITH A CHANGE OF VARIABLES
• Find
FINDING AREA WITH THE LOG RULE
• Find the area of the region bounded by the graph of
the x-axis, and the line
RECOGNIZING QUOTIENT FORMS OF THE LOG RULE
∫ 3 𝑥2+1
𝑥3+𝑥𝑑𝑥
RECOGNIZING QUOTIENT FORMS OF THE LOG RULE
∫ sec2𝑥tan𝑥
𝑑𝑥
RECOGNIZING QUOTIENT FORMS OF THE LOG RULE
∫ 𝑥+1𝑥2+2𝑥
𝑑𝑥
RECOGNIZING QUOTIENT FORMS OF THE LOG RULE
∫ 13 𝑥+2
𝑑𝑥
USING LONG DIVISION BEFORE INTEGRATING
• Find
CHANGE OF VARIABLES WITH THE LOG RULE
• Find
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
USING A TRIG IDENTITY
• Find
INTEGRALS OF THE SIX BASIC TRIGONOMETRIC FUNCTIONS
INTEGRATING TRIGONOMETRIC FUNCTIONS
• Evaluate
FINDING AN AVERAGE VALUE
• Find the average value of on the interval [0, ]
CH 5.2 HOMEWORK
• Pg 338 – 339, #’s: 1 – 41 every other odd, 47, 71, 75 (ignore the direction about Simpson’s rule)