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Substitution
Michael Freeze
MAT 151 UNC Wilmington
Summer 2013
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Section 7.2 :: Substitution
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Substitution Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 - - PowerPoint PPT Presentation
Substitution Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 - - PowerPoint PPT Presentation
Substitution Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 17 Section 7.2 :: Substitution 2 / 17 Substitution Form of the Integral [ f ( x )] n f ( x ) dx , n = 1 Result of Substituting u = f ( x ) n + 1 + C = [ f (
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SLIDE 3
Substitution
Form of the Integral
- [f (x)]n f ′(x) dx, n = −1
Result of Substituting u = f (x)
- un du = un+1
n + 1 + C = [f (x)]n+1 n + 1 + C
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SLIDE 4
Substitution
Form of the Integral f ′(x) f (x) dx Result of Substituting u = f (x) 1 u du = ln |u| + C = ln |f (x)| + C
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SLIDE 5
Substitution
Form of the Integral
- ef (x) f ′(x) dx
Result of Substituting u = f (x)
- eu du = eu + C = ef (x) + C
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SLIDE 6
Substitution Method
Often we can choose u to be one of the following:
1 the quantity under a root or raised to a power 2 the quantity in the denominator 3 the exponent on e
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SLIDE 7
Basic Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- (2x + 3)3 · 2 dx
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SLIDE 8
Basic Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- 5 · (5x − 9)2 dx
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SLIDE 9
Basic Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- (−4x + 1)7 dx
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SLIDE 10
Basic Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
3x2 − 5 4 · 2x dx
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SLIDE 11
Intermediate Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- x + 1
(x2 + 2x − 4)4 dx
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SLIDE 12
Intermediate Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- x2
2x3 + 7 dx
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SLIDE 13
Intermediate Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- x2
3
√ 2x3 + 7 dx
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SLIDE 14
Intermediate Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- x e3x2−5 dx
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SLIDE 15
Advanced Integration by Substitution Examples
General Power Rule for Integration For any real number n = −1,
- [f (x)]n f ′(x) dx = [f (x)]n+1
n + 1 + C. Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
(1 + ln x)2 x dx
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SLIDE 16
Advanced Integration by Substitution Examples
Recall that f ′(x) f (x) dx = ln |f (x)| + C Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- x + 1
x2 + 2x − 4
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SLIDE 17
Advanced Integration by Substitution Examples
Recall that f ′(x) f (x) dx = ln |f (x)| + C Here, u = f (x) and du = f ′(x) dx. Evaluate the indefinite integral.
- e2x
e2x + 3 dx
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