mat 1235 calculus ii section 7.1 integration by parts
TRANSCRIPT
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MAT 1235Calculus II
Section 7.1
Integration By Parts
http://myhome.spu.edu/lauw
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Homework and …
WebAssign 7.1(17 problems, 38 min.)
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Exam II
Be sure to look at the quiz solutions. If you have questions, resolve them
ASAP. Do not wait until the exam day to ask that question.
Get help before it is too late. Exam II is coming up in 8 days.
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Preview
The reverse of the product rule
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Product Rule
dxxgxfxgxfdxxgxf
xgxfdxxgxfdxxgxf
xgxfdxxgxfxgxf
xgxfxgxfxgxfdx
d
)()()()()()(
)()()()()()(
)()()()()()(
)()()()()()(
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Product Rule
dxxgxfxgxfdxxgxf
xgxfdxxgxfdxxgxf
xgxfdxxgxfxgxf
xgxfxgxfxgxfdx
d
)()()()()()(
)()()()()()(
)()()()()()(
)()()()()()(
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Definition of Antiderivatives
dxxgxfxgxfdxxgxf
xgxfdxxgxfdxxgxf
xgxfdxxgxfxgxf
xgxfxgxfxgxfdx
d
)()()()()()(
)()()()()()(
)()()()()()(
)()()()()()(
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Rationale
dxxgxfxgxfdxxgxf )()()()()()(
The integral on the right hand side is easier to evaluate than the one on the left hand side
Easier
Difficult
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Rationale
dxxgxfxgxfdxxgxf )()()()()()(
The integral on the right hand side is easier to evaluate than the one on the left hand side
Easier
Difficult
xdxxxxdxx sinsincos
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Alternative Form
dxxgxfxgxfdxxgxf )()()()()()(
vduuvudv
dxxgdvdxxfdu
xgvxfu
have We
)( ,)(Then
)( ),(Let
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Integration By Parts
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )b b
b
aa a
f x g x dx f x g x f x g x dx
udv uv vdu
f x g x dx f x g x f x g x dx
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Example 1
cosx xdx
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Example 1(Analysis) vduuvudv
In order to use IBP, we need to choose and
One possible choice is xdx dvxu cos ,
cosx xdx
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Example 1(Analysis)
In order to use IBP, we need to choose and
One possible choice is
xdx dvxu cos ,Let
cosx xdx
vduuvudv
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Example 1(Analysis)
Now, we need to find and
xdxx cos
x
xdxvdxdu
xdx dvxu
sin
cos , Then,
cos , Let
vduuvudv
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Example 1(Analysis)
Now, we need to find and
xdxx cos
x
xdxvdxdu
xdx dvxu
sin
cos , Then,
cos , Let
vduuvudv
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Example 1
cos
sin sin
x xdx
x x xdx
x
xdxvdxdu
xdx dvxu
sin
cos , Then,
cos , Let
vduuvudv
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Example 1
cos
sin sin
x xdx
x x xdx
vduuvudv
x
xdxvdxdu
xdx dvxu
sin
cos , Then,
cos , Let
What if?
B
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Example 1 vduuvudv
What if?
2 ,sin Then,
,cos Let 2x
xdxvxdxdu
xdx dvxu
cos
cos
x xdx
x xdx
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Example 2xxe dx
vduuvudv
xxe dx
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Expectations
All supporting steps are required and done on the right side.
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Example 32
1
ln xdx
vduuvudv
2
1
ln xdx
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Example 4
cosxe xdx
vduuvudv
cosxe xdx xe