Evaluate the integral by applying the following theorems and the power rule appropriately. Suppose that F(x) and G(x) are antiderivatives of f(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, S(x) + g(x)] dr = F(x)+ G(x)+ C. (c) An antiderivative of a difference is the difference of the antiderivatives; that is, |Is(x) – g(x)] da = F(x) – G(æ) + C. x"+1 The power rule: x" dx + C,r + -1. r+1 NOTE: Enter the exact answer. 14 3 dy = |+C
Evaluate the integral by applying the following theorems and the power rule appropriately. Suppose that F(x) and G(x) are antiderivatives of f(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, S(x) + g(x)] dr = F(x)+ G(x)+ C. (c) An antiderivative of a difference is the difference of the antiderivatives; that is, |Is(x) – g(x)] da = F(x) – G(æ) + C. x"+1 The power rule: x" dx + C,r + -1. r+1 NOTE: Enter the exact answer. 14 3 dy = |+C
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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Question
![Evaluate the integral by applying the following theorems
and the power rule appropriately.
Suppose that F (x) and G(x) are antiderivatives of f(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,
| ef (x) dæ = cF(x)+ C.
(b) An antiderivative of a sum is the sum of the antiderivatives;
that is,
|[f (x) + g(x)] dæ = F(æ)+ G(w)+ C.
(c) An antiderivative of a difference is the difference of the
antiderivatives; that is,
|S(x) – g(x)| dæ = F(æ) – G(æ) +C.
x"+1
: a" dx = +C,r #-1.
The
power
rule:
r+1
NOTE: Enter the exact answer.
14
dy =
+C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2c33135-357a-40fe-834b-3c5e0761da65%2F830b90b1-c731-47d1-afd4-72c58e59dda6%2F3kyjnrl_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 f(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,
| ef (x) dæ = cF(x)+ C.
(b) An antiderivative of a sum is the sum of the antiderivatives;
that is,
|[f (x) + g(x)] dæ = F(æ)+ G(w)+ C.
(c) An antiderivative of a difference is the difference of the
antiderivatives; that is,
|S(x) – g(x)| dæ = F(æ) – G(æ) +C.
x"+1
: a" dx = +C,r #-1.
The
power
rule:
r+1
NOTE: Enter the exact answer.
14
dy =
+C
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