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)