chapter 5 section 5.2 the natural logarithmic function: integration
TRANSCRIPT
CHAPTER 5SECTION 5.2
THE NATURAL LOGARITHMIC FUNCTION:
INTEGRATION
Theorem 5.5 Log Rule for Integration
1du
uTheorem:
21. dx
x
12.
4 1dx
x
1du
uTheorem:
21. dx
x
ln u C
12.
4 1dx
x
12 dxx
2ln x C
4 1u x 4du dx
1 1
4du
u
1ln
4u C
1ln 4 1
4x C
2
43.
4 1
xdx
x
22
44.
4 1
xdx
x
2
43.
4 1
xdx
x
22
44.
4 1
xdx
x
2
4 8
8 4 1
xdx
x
24 1u x
8du xdx
1 1
2du
u
1ln
2u C
21ln 4 1
2x C
22
4 8
8 4 1
xdx
x
24 1u x 8du xdx
2
1 1
2du
u
21
2u du
11
2u C
1
2C
u
2
1
2 4 1C
x
Using Long Division Before Integrating
• Use of the log rule is often in disguised form
• Do the division on this integrand and alter it's appearance
22 7 3
2
x xdx
x
2
2 11 Remainder 19
2 2 7 3
x
x x x
Using Long Division Before Integrating
• Now take the integral
192 11
2x dx
x
Change of Variables
• Consider
– Then u = x – 1 and du = dx– But x = u + 1 and x – 2 = u – 1
• So we have
– Finish the integration
3
2
1
x xdx
x
3
1 1u udu
u
Guidelines for Integration
2
25.
1
xdx
x
16.
lndx
x x
2
25.
1
xdx
x
16.
lndx
x x
1u x du dx
22
xdu
u
1u x
2
12u
duu
21
2 u duu
12 ln u u C
22ln 1
1x C
x
lnu x1
du dxx
1du
u ln u C ln lnx C
Integrals of the Six Basic Trigonometric Functions
Explanations:
tan xdx cot xdx
sec xdx csc xdx
Explanations:
tan xdx cot xdx sin
cos
xdxxcosu x
sindu xdx
1du
u ln u C
ln cos x C cos
sin
xdx
xsinu xcosdu xdx
1du
u ln u C
ln sinx C
sec xdx sec tan
se
sec
1 c tan
x x
xd
x
xx
sec tanu x x
2sec sec tan
sec tan
x x xdx
x x
2sec sec tandu x x x dx 1du
u ln u C
ln sec tanx x C csc xdx csc cot
cs
csc
1 c cot
x x
xd
x
xx
csc cotu x x
2csc csc cot
csc cot
x x xdx
x x
2csc csc cotdu x x x dx 1du
u ln u C
ln csc cotx x C
4
2
0
1 tan x dx
4
2
0
1 tan x dx
4
2
0
sec x dx
4
0
sec xdx
4
0ln sec tanx x
4 4ln sec tan ln sec 0 tan0
ln 2 1 ln 1 0
ln 2 1 0
ln 2 1