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

⎡⎣

⎤⎦