c. Observe that the examples in (b) work nicely because of the derivati you were asked to calculate in (a). Each integrand in (b) is precisely t result of differentiating one of the products of basic functions found (a). To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we
c. Observe that the examples in (b) work nicely because of the derivati you were asked to calculate in (a). Each integrand in (b) is precisely t result of differentiating one of the products of basic functions found (a). To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Only need help with part c please, thank you!
![V.
[₁ 1+ln(x) dx
c. Observe that the examples in (b) work nicely because of the derivatives
you were asked to calculate in (a). Each integrand in (b) is precisely the
result of differentiating one of the products of basic functions found in
(a). To see what happens when an integrand is still a product but not
necessarily the result of differentiating an elementary product, we
consider how to evaluate
i. First, observe that
[xc
x cos(x) dx.
d
[x sin(x)] = x cos(x) + sin(x).
dx
Integrating both sides indefinitely and using the fact that the
integral of a sum is the sum of the integrals, we find that
( * sin(x)]
dx
dx
= [x cos(x) dx + [ sin(x) dx.
In this last equation, evaluate the indefinite integral on the left side
as well as the rightmost indefinite integral on the right.
ii. In the most recent equation from (i.), solve the equation for the
expression fx cos(x) dx.
iii. For which product of basic functions have you now found the
antiderivative?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce3589f0-5608-4fb5-a11c-ce1805f2c8e6%2F67d1f9e6-d00c-469e-b535-0851e8846a7c%2Fcdu6de_processed.png&w=3840&q=75)
Transcribed Image Text:V.
[₁ 1+ln(x) dx
c. Observe that the examples in (b) work nicely because of the derivatives
you were asked to calculate in (a). Each integrand in (b) is precisely the
result of differentiating one of the products of basic functions found in
(a). To see what happens when an integrand is still a product but not
necessarily the result of differentiating an elementary product, we
consider how to evaluate
i. First, observe that
[xc
x cos(x) dx.
d
[x sin(x)] = x cos(x) + sin(x).
dx
Integrating both sides indefinitely and using the fact that the
integral of a sum is the sum of the integrals, we find that
( * sin(x)]
dx
dx
= [x cos(x) dx + [ sin(x) dx.
In this last equation, evaluate the indefinite integral on the left side
as well as the rightmost indefinite integral on the right.
ii. In the most recent equation from (i.), solve the equation for the
expression fx cos(x) dx.
iii. For which product of basic functions have you now found the
antiderivative?
![Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and
studied how it is employed to differentiate a product of two functions. In
particular, recall that if f and g are differentiable functions of x, then
a. For each of the following functions, use the Product Rule to find the
function's derivative. Be sure to label each derivative by name (e.g., the
derivative of g(x) should be labeled g'(x)).
i.
d
-[ƒf(x) · g(x)] = f(x) · g'(x) + g(x) · ƒ'(x).
dx
i. g(x) = x sin(x)
iii. p(x) = x ln(x)
v. r(x) = e* sin(x)
b. Use your work in (a) to help you evaluate the following indefinite
integrals. Use differentiation to check your work.
iii.
V.
xe te da
[2x
[1+
1 + ln(x) dx
ii. h(x):
=xe*
iv. q(x) = x² cos(x)
2x cos(x) — x² sin(x) dx
ii.
iv.
e* (sin(x) + cos(x)) dx
[x
x cos(x) + sin(x) dx
c. Observe that the examples in (b) work nicely because of the derivatives
you were asked to calculate in (a). Each integrand in (b) is precisely the
result of differentiating one of the products of basic functions found in
(a). To see what happens when an integrand is still a product but not
necessarily the result of differentiating an elementary product, we
consider how to evaluate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce3589f0-5608-4fb5-a11c-ce1805f2c8e6%2F67d1f9e6-d00c-469e-b535-0851e8846a7c%2Fekljvs8_processed.png&w=3840&q=75)
Transcribed Image Text:Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and
studied how it is employed to differentiate a product of two functions. In
particular, recall that if f and g are differentiable functions of x, then
a. For each of the following functions, use the Product Rule to find the
function's derivative. Be sure to label each derivative by name (e.g., the
derivative of g(x) should be labeled g'(x)).
i.
d
-[ƒf(x) · g(x)] = f(x) · g'(x) + g(x) · ƒ'(x).
dx
i. g(x) = x sin(x)
iii. p(x) = x ln(x)
v. r(x) = e* sin(x)
b. Use your work in (a) to help you evaluate the following indefinite
integrals. Use differentiation to check your work.
iii.
V.
xe te da
[2x
[1+
1 + ln(x) dx
ii. h(x):
=xe*
iv. q(x) = x² cos(x)
2x cos(x) — x² sin(x) dx
ii.
iv.
e* (sin(x) + cos(x)) dx
[x
x cos(x) + sin(x) dx
c. Observe that the examples in (b) work nicely because of the derivatives
you were asked to calculate in (a). Each integrand in (b) is precisely the
result of differentiating one of the products of basic functions found in
(a). To see what happens when an integrand is still a product but not
necessarily the result of differentiating an elementary product, we
consider how to evaluate
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