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Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)
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Transcript of Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)
![Page 1: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/1.jpg)
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
![Page 2: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/2.jpg)
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Cxex ln73
![Page 3: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/3.jpg)
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Cxex ln73
Cxx
3
sin
3
2 12
3
![Page 4: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/4.jpg)
Integration by Parts
Section 8.2
Objective: To integrate problems without a u-substitution
![Page 5: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/5.jpg)
Integration by Parts• When integrating the product of two functions, we
often use a u-substitution to make the problem easier to integrate. Sometimes this is not possible. We need another way to solve such problems.
)()( xgxf
![Page 6: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/6.jpg)
Integration by Parts• As a first step, we will take the derivative of )()( xgxf
![Page 7: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/7.jpg)
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
![Page 8: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/8.jpg)
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
)()()()()()( // xfxgxgxfxgxfdx
d
![Page 9: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/9.jpg)
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
)()()()()()( // xfxgxgxfxgxfdx
d
)()()()()()( // xfxgxgxfxgxf
![Page 10: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/10.jpg)
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
)()()()()()( // xfxgxgxfxgxfdx
d
)()()()()()( // xfxgxgxfxgxf
)()()()()()( // xgxfxfxgxgxf
![Page 11: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/11.jpg)
Integration by Parts• Now lets make some substitutions to make this easier
to apply.)(xgv )(xfu
)()()()()()( // xgxfxfxgxgxf
)(/ xgdv )(/ xfdu
udvvduuv
![Page 12: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/12.jpg)
Integration by Parts• This is the way we will look at these problems.
• The two functions in the original problem we are integrating are u and dv. The first thing we will do is to choose one function for u and the other function will be dv.
)(xgv )(xfu
)(/ xgdv )(/ xfdu udvvduuv
![Page 13: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/13.jpg)
Example 1• Use integration by parts to evaluate xdxx cos
![Page 14: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/14.jpg)
Example 1• Use integration by parts to evaluate
xu xdxdv cos
xdxx cos
![Page 15: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/15.jpg)
Example 1• Use integration by parts to evaluate
xv sin
xu xdxdv cos
dxdu
xdxx cos
![Page 16: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/16.jpg)
Example 1• Use integration by parts to evaluate
xv sin
xu xdxdv cos
dxdu
xdxx cos
xdxxxxdxx sinsincos
![Page 17: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/17.jpg)
Example 1• Use integration by parts to evaluate
xv sin
xu xdxdv cos
dxdu
xdxx cos
xdxxxxdxx sinsincos
Cxxxxdxx cossincos
![Page 18: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/18.jpg)
Guidelines
• The first step in integration by parts is to choose u and dv to obtain a new integral that is easier to evaluate than the original. In general, there are no hard and fast rules for doing this; it is mainly a matter of experience that comes from lots of practice.
![Page 19: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/19.jpg)
Guidelines
• There is a useful strategy that may help when choosing u and dv. When the integrand is a product of two functions from different categories in the following list , you should make u the function whose category occurs earlier in the list.
• Logarithmic, Inverse Trig, Algebraic, Trig, Exponential
• The acronym LIATE may help you remember the order.
![Page 20: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/20.jpg)
Guidelines
• If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.
![Page 21: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/21.jpg)
Guidelines
• If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.
xdxx cosxu cos
xdxdu sin
2
2xv
xdxdv
xdxx
xx
xdxx sin2
cos2
cos22
![Page 22: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/22.jpg)
Guidelines
• Since the new integral is harder than the original, we made the wrong choice.
xdxx cosxu cos
xdxdu sin
2
2xv
xdxdv
xdxx
xx
xdxx sin2
cos2
cos22
![Page 23: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/23.jpg)
Example 2• Use integration by parts to evaluate dxxex
![Page 24: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/24.jpg)
Example 2• Use integration by parts to evaluate
xu dxedv x
dxxex
![Page 25: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/25.jpg)
Example 2• Use integration by parts to evaluate
xev
xu dxedv x
dxdu
dxxex
![Page 26: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/26.jpg)
Example 2• Use integration by parts to evaluate
xev
xu dxedv x
dxdu
dxxex
dxexedxxe xxx
![Page 27: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/27.jpg)
Example 2• Use integration by parts to evaluate
xev
xu dxedv x
dxdu
dxxex
dxexedxxe xxx
Cexedxxe xxx
![Page 28: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/28.jpg)
Example 3 (S):• Use integration by parts to evaluate xdxln
![Page 29: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/29.jpg)
Example 3• Use integration by parts to evaluate
xu ln dxdv
xdxln
![Page 30: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/30.jpg)
Example 3• Use integration by parts to evaluate
xv
xu ln dxdv
dxx
du1
xdxln
![Page 31: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/31.jpg)
Example 3• Use integration by parts to evaluate
xv
xu ln dxdv
dxx
du1
xdxln
dxxxxdx lnln
![Page 32: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/32.jpg)
Example 3• Use integration by parts to evaluate
xv
xu ln dxdv
dxx
du1
xdxln
dxxxxdx lnln
Cxxxxdx lnln
![Page 33: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/33.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate dxex x2
![Page 34: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/34.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate
2xu dxedv x dxex x2
![Page 35: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/35.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
![Page 36: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/36.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222
![Page 37: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/37.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxedv x
![Page 38: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/38.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxdu xev
dxedv x
![Page 39: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/39.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxdu xev
dxedv x
dxexeexdxex xxxx 222
![Page 40: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/40.jpg)
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxdu xev
dxedv x
dxexeexdxex xxxx 222
Cexeexdxex xxxx 2222
![Page 41: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/41.jpg)
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
![Page 42: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/42.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
dxdv
![Page 43: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/43.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
![Page 44: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/44.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
![Page 45: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/45.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
![Page 46: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/46.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu xdxdu 2
![Page 47: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/47.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
dxx
du
2
xdxdu 2
![Page 48: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/48.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
dxx
du
2
xdxdu 2
u
duxxdxx2
1tan)(tan 1
1
0
1
![Page 49: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/49.jpg)
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
dxx
du
2
xdxdu 2
u
duxxdxx2
1tan)(tan 1
1
0
1
10211
0
1 )1ln(2
1tan)(tan xxxdxx
![Page 50: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/50.jpg)
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
10211
0
1 )1ln(2
1tan)(tan xxxdxx
![Page 51: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/51.jpg)
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
10211
0
1 )1ln(2
1tan)(tan xxxdxx
)01ln(2
10tan0)11ln(
2
11tan1)(tan 2121
1
0
1 dxx
![Page 52: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/52.jpg)
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
)1ln(2
1tan)(tan 21
1
0
1 xxxdxx
)01ln(2
10tan0)11ln(
2
11tan1)(tan 2121
1
0
1 dxx
002ln2
1
4)(tan
1
0
1 dxx
![Page 53: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/53.jpg)
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
)1ln(2
1tan)(tan 21
1
0
1 xxxdxx
)01ln(2
10tan0)11ln(
2
11tan1)(tan 2121
1
0
1 dxx
2ln4
002ln2
1
4)(tan
1
0
1 dxx
![Page 54: Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649dcf5503460f94ac3b7b/html5/thumbnails/54.jpg)
Homework:Page 520
# 3-9 odd, 15, 25, 29, 31, 37