MATH 131-503 Fall 2015
c
Wen
Liu
5.5
5.5 The Substitution Rule
The Substitution Rule:
• If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I,
then
Z
Z
0
f (g(x))g (x)dx = f (u)du
• If g 0 is continuous on [a, b] and f is continuous on the range of u = g(x), then
Z
b
0
Z
g(b)
f (g(x))g (x)dx =
a
f (u)du
g(a)
Examples:
Z
1.
e−10x dx, u = −10x
Z
2.
dt
, u = 1 − 3t
(1 − 3t)3
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MATH 131-503 Fall 2015
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sin(ln(43x))
dx
x
Z
e9x
dx
e9x + 2
3.
4.
Z
5.
5.5
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Liu
a
3x7 (a + 2x8 )4 dx for some constant a
0
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MATH 131-503 Fall 2015
Z
e64
6.
e25
5.5
c
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Liu
dx
√
x ln x
7. A bacteria population starts with 350 bacteria and grows at a rate of r(t) = 450.268e1.12567t
bacteria per hour. How many bacteria will there be after three hours?
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