Chapter6: Exponential And Logarithmic Functions
Section6.7: Exponential And Logarithmic Models
Problem 27SE: Prove that bx=exln(b) for positive b1 .
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![Evaluate the integral by applying the following theorems
and the power rule appropriately.
Suppose that F(x) and G(x) are antiderivatives of ƒf(x) and g(x)
respectively, and that c is a constant. Then:
(a) A constant factor can be moved through an integral sign; that is,
[cf(x) dx = cF(x) + C.
(b) An antiderivative of a sum is the sum of the antiderivatives;
that is,
[ {f(x) + g(x)] dx = F(x) +G(x) + C.
(c) An antiderivative of a difference is the difference of the
antiderivatives; that is,
[{f(x) − g(x)] dx = F(x) − G(x) + C.
Xr+1
r+1
J
NOTE: Enter the exact answer.
The power rule:
x" dx
[[112+] dr = [
dx
8x³
+ C, r = -1.
+C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F449aec2f-1d59-4dc0-a1f8-18a413b02ad6%2F2d5f13f7-377f-43d3-858a-f26c2f2efb65%2Fejxjzp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Evaluate the integral by applying the following theorems
and the power rule appropriately.
Suppose that F(x) and G(x) are antiderivatives of ƒf(x) and g(x)
respectively, and that c is a constant. Then:
(a) A constant factor can be moved through an integral sign; that is,
[cf(x) dx = cF(x) + C.
(b) An antiderivative of a sum is the sum of the antiderivatives;
that is,
[ {f(x) + g(x)] dx = F(x) +G(x) + C.
(c) An antiderivative of a difference is the difference of the
antiderivatives; that is,
[{f(x) − g(x)] dx = F(x) − G(x) + C.
Xr+1
r+1
J
NOTE: Enter the exact answer.
The power rule:
x" dx
[[112+] dr = [
dx
8x³
+ C, r = -1.
+C
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