EC PPT Integration the Antiderivative C represents Constant Raise the power Guess Check Answer Ex 2X dx Guess X 2 Check 2X Answer X 2 C Off by a constant good ID: 792368
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Slide1
Integration
Jami Wang
Period 3
EC PPT
Slide2Integration – the Antiderivative
C represents ConstantRaise the power
Guess. Check. Answer.
Ex:
∫
2X
dx
Guess: X
2
Check: 2X
Answer : X
2
+ C
Off by a constant = good
Off by a variable = new guess
Slide3Integration
– the Antiderivative (continued)
When power raised = zero, use
ln
Ex: ∫ 2/X
dx
( ∫ 2X
-1
dx
)
Guess:
2
lnX
Check: (
2
) 1/x
Answer: 2lnx + C
or
ln
X
2
+ C
Slide4Most Complicated Rule
Start out with the complicated part and ignore the easy part
Can’t decide the most complicated one – choose one
Ex: ∫ X (X
2
-
1)
9
Guess:
1/20
(
X
2
-
1)
10
Check: 10 (
X
2
-
1)
9
(2X)
Answer:
1/20
(X
2
- 1)
10
+ C
Slide5Ex2: ∫ (
ln x6 / x) dx
Guess:
1/7
(
lnx
)
7
Check: (
lnx
)
6 (1/x)Answer: 1/7 (lnx) 7 + C
Most Complicated Rule (continued)
Slide6Definite Integrals
Ex
:
3
0
∫ √ (y+1)
dy
3
∫
0
√ (y+1)
1/2
dy Guess: 2/3 (y+1)
3/2
Check
: 3/2 (y+1) 1/2substitution then subtractionBiggest number goes first2/3 (y+1) 3/2 30 = 2/3 (4) 3/2 – 2/3 (1) 3/2 *use x values to find y values Answer: 14/3