CalculusVolume2-SASG-01-05
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 1
Integration
1.5. Substitution
Section Exercises
255.
If when reversing the chain rule, should you take or ?
Answer: In the following exercises, verify each identity using differentiation. Then, using the indicated u
-substitution, identify f
such that the integral takes the form .
257.
Answer: 259.
; Answer: ; In the following exercises, find the antiderivative using the indicated substitution.
261.
; Answer: 263.
; Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
265.
; Answer: 267.
; Answer: 269.
; (
Hint:
)
Answer: In the following exercises, use a suitable change of variables to determine the indefinite integral.
271.
Answer: 273.
Answer: 275.
Answer: 277.
Answer: 279.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
281.
Answer: 283.
Answer: 285.
Answer: 287.
Answer: In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer.
289.
[T] over Answer: The exact area is 291.
[T] over Answer: … The exact area is 0.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
In the following exercises, use a change of variables to evaluate the definite integral.
293.
Answer: , , 295.
Answer: , , 297.
Answer: , , In the following exercises, evaluate the indefinite integral with constant using u
-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C
that would need to be added to the antiderivative to make it equal to the definite integral , with a
the left endpoint of the given interval.
299.
[T] on Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
The antiderivative is . Since the antiderivative is not continuous at , one cannot find a value of C
that would make work as a definite integral.
301.
[T] over Answer: The antiderivative is . You should take so that .
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
303.
[T] over Answer: The antiderivative is . One should take .
305.
Is the substitution in the definite integral okay? If not, why not?
Answer: No, because the integrand is discontinuous at .
In the following exercises, use a change of variables to show that each definite integral is equal to zero.
307.
Answer: ; the integral becomes .
309.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: ; the integral becomes .
OpenStax Calculus Volume 2
Student Answer and Solution Guide
311.
Answer: ; the integral becomes since the integrand is odd.
313.
Show that the average value of over an interval is the same as the average value of over the interval for .
Answer: Setting and gets you .
315.
Find the area under the graph of between and , where
and is fixed. Evaluate the limit as .
Answer: As the limit is if , and the limit diverges to +∞ if .
317.
The area of the top half of an ellipse with a major axis that is the x
-axis from to a
and with a minor axis that is the y
-axis from to b
can be written as . Use the substitution to express this area in terms of an integral of a trigonometric function.
You do not need to compute the integral.
Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
319.
[T] The following graph is of a function of the form . Estimate the coefficients a
and b
and the frequency parameters n
and m
. Use these estimates to approximate .
Answer: ; This file is copyright 2016, Rice University. All Rights Reserved.