the natural logarithmic function: integration (5.2)
DESCRIPTION
The Natural Logarithmic Function: Integration (5.2). February 26th, 2013. I. Log Rule for Integration. Thm. 5.5: Log Rule for Integration: Let u be a differentiable function of x. 1. 2. Ex. 1: Find each indefinite integral. a. b. c. - PowerPoint PPT PresentationTRANSCRIPT
The Natural Logarithmic Function: Integration (5.2)The Natural Logarithmic
Function: Integration (5.2)February 26th, 2013February 26th, 2013
I. Log Rule for Integration
Thm. 5.5: Log Rule for Integration: Let u be a differentiable function of x.
1.
2.
1
xdx =ln x +C∫1
udu =lnu +C∫
⇒u '
udx = ln u +C∫
Ex. 1: Find each indefinite integral.
a.
b.
c.
4
xdx∫4
2x−1dx∫
x
x2 +6dx∫
Ex. 2: Find the area of the region bounded by the graphs of the equations , x=1, x=4, and y=0.
y=x2 + 4
x
Ex. 3: Find .x4 + x−4
x2 +2dx∫
Ex. 4: Find .3x
(x−2)2dx∫
*Guidelines for Integration1. Know the 12 basic integration formulas you’ve already learned: the power rule, the log rule, and the 10 trigonometric rules.
2. Try to recognize which of those formulas best matches the integrand, and choose u accordingly.
3. If nothing fits, try to manipulate the integrand using algebra or trigonometric identities.
Ex. 5: Solve the differential equation .
dy
dx=
1xln x3( )
II. Integrals of the Trigonometric Functions
sinudu =−cosu+C∫cosudu =sinu+C∫tanudu =−lncosu +C∫cotudu =lnsinu +C∫secudu =lnsecu+ tanu +C∫cscudu =−lncscu+cotu +C∫
Ex. 6: Find .sec2 2θ −10
π6
∫ dθ
Ex. 7: Find the average value of f(x)=csc x on the interval .π
4,3π4
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