(c) 2014 joan s kessler distancemath.com 11 integration by parts
TRANSCRIPT
(c) 2014 joan s kessler distancemath.com
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Integration By Parts
sin( )x x dx
2 xx e dxln x dx
cos xe x dx
(c) 2014 joan s kessler distancemath.com
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Suppose we want to integrate this function.
Up until now we have no way of doing this.
x, and sin(x) seem totally unrelated.
If u(x) is a function and v(x) is another function we seem to have
It almost seems like the reverse of the product rule. Let’s explore the product rule.
sin( )x x dx
u v dx
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Integration By Parts
Start with the product rule:
d dv duuv u v
dx dx dx
d uv u dv v du
u dv d uv v du
u dv d uv v du
u dv d uv v du
u dv uv v du
This is the Integration by Parts formula.
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u dv uv v du
The Integration by Parts formula is a “product The Integration by Parts formula is a “product rule” for integration.rule” for integration.
u differentiates to zero (usually).
dv is easy to integrate.
Choose u in this order: LIPET
Logs, Inverse trig, Polynomial, Exponential, Trig
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Example 1:
cos x x dxpolynomial factor u x
du dx
cos dv x dx
sinv x
u dv uv v du u dv uv v du LIPETLIPET
sin cosx x x C
u v v du
sin sin x x x dx
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Example 2:
ln x dxlogarithmic factor lnu x
1du dx
x
dv dx
v x
u dv uv v du LIPET
lnx x x C
1ln x x x dx
x
u v v du
ln x dx
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This is still a product, so we need to use integration by parts againagain.
Example 3:2 xx e dx
u dv uv v du LIPET
2u x xdv e dx
2 du x dx xv e u v v du
2 2 x xx e e x dx 2 2 x xx e xe dx u x xdv e dx
du dx xv e 2 2x x xx e xe e dx
2 2 2x x xx e xe e C 2 xx e dx
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Example 4:
cos xe x dxLIPET
xu e sin dv x dx xdu e dx cosv x
u v v du sin sinx xe x x e dx
sin cos cos x x xe x e x x e dx
xu e cos dv x dx xdu e dx sinv x
sin cos cos x x xe x e x e x dx
This is the expression we started with!
uv v du
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Example 5:
cos xe x dxLIPET
u v v du
cos xe x dx 2 cos sin cosx x xe x dx e x e x
sin coscos
2
x xx e x e x
e x dx C
sin sinx xe x x e dx
xu e sin dv x dxxdu e dx cosv x
xu e cos dv x dx xdu e dx sinv x
sin cos cos x x xe x e x e x dx
sin cos cos x x xe x e x x e dx
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Example 5 :
cos xe x dx u v v du
This is called “solving for the unknown integral.”
It works when both factors integrate and differentiate forever.
cos xe x dx 2 cos sin cosx x xe x dx e x e x
sin coscos
2
x xx e x e x
e x dx C
sin sinx xe x x e dx
sin cos cos x x xe x e x e x dx
sin cos cos x x xe x e x x e dx
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More integration by Parts Ex 6.More integration by Parts Ex 6.
u v v du
2
3arcsin 3
1- 9x x x dx
x
2 -1/2arcsin 3 3 (1 9 ) x x x x dx
arcsin 3x dx Let u = arcsin3x dv = dx
v = x2
3
1 9du dx
x
2 1/ 21arcsin 3 (1 9 )
3x x x C
-18 -118
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More integration by Parts Ex. 7More integration by Parts Ex. 7
u v v du
22 2
1 1(2x)
2( 5) 2( 5)x dx
x x
22
2
1ln( 5)
2( 5) 2
xx C
x
3
2 2( 5)
xdx
x Let u = x2 du = 2x dx
2 2( 5)
xdv dx
x
2
1
2( 5)v
x
Formdu
u
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A Shortcut: Tabular Integration
Tabular integration works for integrals of the form:
f x g x dxwhere:
Differentiates to zero in several steps.
Integrates repeatedly.
u dv
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2 xx e dx de .& v rif x integ & ralsg x
2x
2x
20
xexe
xexe
2 xx e dx 2 xx e 2 xxe 2 xe C
Compare this with the same problem done the other way:
Alternate signs
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Same Example :2 xx e dx
u dv uv v du LIPET
2u x xdv e dx
2 du x dx xv e u v v du
2 2 x xx e e x dx 2 2 x xx e xe dx u x xdv e dx
du dx xv e 2 2x x xx e xe e dx
2 2 2x x xx e xe e C This is easier and quicker to do with tabular integration!
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2 2 2x x xx e xe e C
2( 2 2)xe x x C
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3 sin x x dx
3x
23x
6x
6
sin x
cos x
sin xcos x
0
sin x
3 cosx x 2 3 sinx x 6 cosx x 6sin x + C
de .& v rif x integ & ralsg x
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5 sinx dx5
4
3
2
sin
5 cos
20 sin
60 cos
120 sin
120 cos
0 sin
x x
x x
x x
x x
x x
x
x
5 5 4 3 2sin cos 5 sin 20 cos 60 sin 120 cos 120sinx dx x x x x x x x x x x x C
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5(3 2) xx e dx
3 2x
3
5xe5
5
xe
5
25
xe
0
5(3 2)
5
xx e 53
25
xe
5 53 7
5 25
x xxe eC + C
de .& v rif x integ & ralsg x
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5 53 7
5 25
x xxe eC
51 (15 7)
25xe x C
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Try
3 cos 2x xdx
3 2xx e dx
32 2( 2)x x dx
2 3 214 6 6 3
8xe x x x C
3 214 sin 2 6 cos 2 6 sin 2 3cos 2
8x x x x x x x C
5
222
( 2) 35 40 32315
x x x C
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3 2xx e dx3 2
2 2
2
2
2
13
21
641
681
016
x
x
x
x
x
x e
x e
x e
e
e
++
--
++
--
++
2 3 214 6 6 3
8xe x x x C
3 2 2 2 2 22 3 3 3
2 4 4 8
x x xxx x e xe x e
e C
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3 cos 2x xdx de .& v rif x integ & ralsg x
3 214 sin 2 6 cos 2 6 sin 2 3cos 2
8x x x x x x x C
3
2
cos 2
13 sin 2
21
6 cos 241
6 sin 281
0 cos 216
x x
x x
x x
x
x
3 2sin 2 3 cos 2 3 sin 2 3cos 2
2 4 4 8
x x x x x x xC ++
--
++
--
++
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32 2( 2)x x dx
5
222
( 2) 35 40 32315
x x x C
32 2
5
2
7
2
9
2
( 2)
22 ( 2)
5
42 ( 2)
35
80 ( 2)
315
x x
x x
x
x
++
--
++
--
Homework
Assignment
25