4.3.1 Find the indefinite integrals of the following functions. ...
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4.3.2 Find the indefinite integrals of the following functions. ...
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4.3.3 Find the indefinite integrals of the following functions. ...
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4.3.4 Find the indefinite integrals of the following functions. ...
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4.3.5 Find the indefinite integrals of the following functions. 2 sin(x) + 3 cos(x)
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4.3.6 Find the indefinite integrals of the following functions. ...
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4.3.7
Use substitution, if needed, to find the indefinite integrals of the
following functions. Check with the chain rule. ...
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4.3.8 Use substitution, if needed, to find the indefinite integrals of the following functions. Check with the chain rule. cos[2π(x − 2)]
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4.3.9
Use substitution, if needed, to find the indefinite integrals of the
following functions. Check with the chain rule. ...
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4.3.10
Use substitution, if needed, to find the indefinite integrals of the
following functions. Check with the chain rule. ...
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4.3.11
Use substitution, if needed, to find the indefinite integrals of the
following functions. Check with the chain rule. ...
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4.3.12
Use substitution, if needed, to find the indefinite integrals of the
following functions. Check with the chain rule. ...
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4.3.13 Use substitution to find the indefinite integrals of the following functions. ...
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4.3.14 Use substitution to find the indefinite integrals of the following functions. ...
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4.3.15 Use substitution to find the indefinite integrals of the following functions. ...
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4.3.16 Use substitution to find the indefinite integrals of the following functions. ...
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4.3.17 Use substitution to find the indefinite integrals of the following functions. tan(θ) (write it as ... use a substitution for the denominator.)
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4.3.18 Use substitution to find the indefinite integrals of the following functions. ...
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4.3.19 Use substitution to find the indefinite integrals of the following functions. ...
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4.3.20 Use substitution to find the indefinite integrals of the following functions. ...
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4.3.21 Use integration by parts to evaluate the following. Check your answer by taking the derivative. ...
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4.3.22 Use integration by parts to evaluate the following. Check your answer by taking the derivative. ...
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4.3.23 Use integration by parts to evaluate the following. Check your answer by taking the derivative. ...
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4.3.24 Use integration by parts to evaluate the following. Check your answer by taking the derivative. ...
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4.3.25 Use integration by partial fractions to compute the following indefinite integrals. ...
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4.3.26 Use integration by partial fractions to compute the following indefinite integrals. ...
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4.3.27
Integration by substitution, in combination with trigonometric
identities, can be used to integrate some surprising functions.
Substitute x = tan(θ) to integrate ...
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4.3.28
Integration by substitution, in combination with trigonometric
identities, can be used to integrate some surprising functions.
Substitute x = sin(θ) to integrate ...
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4.3.29
Integration by parts along with substitution can be used to integrate
some of the inverse trigonometric functions. The result of Section
2.10, Exercise 25 gives the derivative of ...Use integration by parts
and a substitution to find ...
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4.3.30
Integration by parts along with substitution can be used to integrate
some of the inverse trigonometric functions. The result of Section
2.10, Exercise 23 gives the derivative of ...Use integration by parts
and a substitution to find ...
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4.3.31 In Examples 4.3.9 and 4.3.10, we chose the constant c =0 when finding v(x). Follow the steps for integration by parts, but leave c as an arbitrary constant. Do you get the same answer? Find the indefinite integral ...as in Example 4.3.9.
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4.3.32 In Examples 4.3.9 and 4.3.10, we chose the constant c =0 when finding v(x). Follow the steps for integration by parts, but leave c as an arbitrary constant. Do you get the same answer? Find the indefinite integral ...as in Example 4.3.10.
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4.3.33
Sometimes integrating by parts seems to lead in a circle, but the
answer can still be found. Try the following. Find the indefinite
integral ...using integration by parts twice
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4.3.34
Sometimes integrating by parts seems to lead in a circle, but the
answer can still be found. Try the following. Find the indefinite
integral ... using integration by parts.
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4.3.35 When
a function has a well-behaved Taylor series, we can find a Taylor
series for some integrals by integrating term by term (proving that
these integrals converge to the correct answer requires methods from
more advanced courses). Integrate the Taylor series for ... term by term, and find the value of the constant for which the integral matches the function ... .
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4.3.36
When
a function has a well-behaved Taylor series, we can find a Taylor
series for some integrals by integrating term by term (proving that
these integrals converge to the correct answer requires methods from
more advanced courses). Integrate the Taylor series for ...term by
term, and check if the answer matches the Taylor series for − ln(1 − x) (use the results of Section 3.7, Exercise 32).
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4.3.37 When
a function has a well-behaved Taylor series, we can find a Taylor
series for some integrals by integrating term by term (proving that
these integrals converge to the correct answer requires methods from
more advanced courses). Integrate the Taylor series for ...term by term. Does this look like a familiar series?
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4.3.38 When
a function has a well-behaved Taylor series, we can find a Taylor
series for some integrals by integrating term by term (proving that
these integrals converge to the correct answer requires methods from
more advanced courses). Integrate the Taylor series for ...term by term. Does this look like a familiar series?
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4.3.39 Integration
by partial fractions works on many more cases than presented in the
main text. We here look at functions with a linear function, rather than
a constant, as the numerator . Find the integral of ...
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4.3.40 Integration
by partial fractions works on many more cases than presented in the
main text. We here look at functions with a linear function, rather than
a constant, as the numerator . Find the integral of ...
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4.3.41 The following differential equations for the production of a chemical P share the properties that the rate of change at t = 0 is 5.0, and the limit as t →∞ is 0. For each, find the solution starting from the initial condition P(0)= 0, sketch the solution, and say what happens to P(t) as t →∞. Compute P(10) and P(100). Why do some increase to infinity while others do not? ...
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4.3.42 The following differential equations for the production of a chemical P share the properties that the rate of change at t = 0 is 5.0, and the limit as t →∞ is 0. For each, find the solution starting from the initial condition P(0)= 0, sketch the solution, and say what happens to P(t) as t →∞. Compute P(10) and P(100). Why do some increase to infinity while others do not? ...
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4.3.43 The following differential equations for the production of a chemical P share the properties that the rate of change at t = 0 is 5.0, and the limit as t →∞ is 0. For each, find the solution starting from the initial condition P(0)= 0, sketch the solution, and say what happens to P(t) as t →∞. Compute P(10) and P(100). Why do some increase to infinity while others do not? ...
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4.3.44 The following differential equations for the production of a chemical P share the properties that the rate of change at t = 0 is 5.0, and the limit as t →∞ is 0. For each, find the solution starting from the initial condition P(0)= 0, sketch the solution, and say what happens to P(t) as t →∞. Compute P(10) and P(100). Why do some increase to infinity while others do not? ...
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4.3.45 Use integration by parts by find solutions of the following differential equations. Suppose the mass M of a toad grows according to the differential equation ...with M(0)= 0. When does this toad grow fastest? Find M(1). How much larger would the toad be if it always grew at the maximum rate?
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4.3.46 Use integration by parts by find solutions of the following differential equations. Suppose the mass W of a worm grows according to the differential equation ...with W(0) = 0. When does this worm grow fastest? Find W(2). How much larger would the worm be if it always grew at the maximum rate?
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4.3.47 The
following problems give the parameters for Walleye in a variety of
locations. For each location, the differential equation has the form ... Find
a. The solution of the differential equation if L(0) = 0.
b. Find the limit of size as t approaches infinity.
c. Assume that all walleye mature at 45 cm in length. How old are these walleye when they mature?
d. Graph the size and compare with Ontario walleye (Figure 4.3.3). (Figure 4.3.3). ... In Texas, where α = 64.3 and β = 1.19.
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4.3.48 equation has the form ... Find
a. The solution of the differential equation if L(0) = 0.
b. Find the limit of size as t approaches infinity.
c. Assume that all walleye mature at 45 cm in length. How old are these walleye when they mature?
d. Graph the size and compare with Ontario walleye (Figure 4.3.3). (Figure 4.3.3). ... In Saskatchewan, where α = 6.48 and β = 0.06.
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4.3.49 The population of lemmings L(t) at the top of a cliff is increasing with the given formula. However, lemmings leap off the cliff at a rate equal to 0.1L and pile up at the bottom.
a. Write a pure-time differential equation for the number of lemmings B(t) piled up at the bottom of the clif
f.
b. Solve using the initial condition B(0) = 0.
c. Graph the number of lemmings at the top and the number at the bottom of the clif
f.
d. Find the limit of the ratio ...at t approaches infinity. ...
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4.3.50 The population of lemmings L(t) at the top of a cliff is increasing with the given formula. However, lemmings leap off the cliff at a rate equal to 0.1L and pile up at the bottom.
a. Write a pure-time differential equation for the number of lemmings B(t) piled up at the bottom of the clif
f.
b. Solve using the initial condition B(0) = 0.
c. Graph the number of lemmings at the top and the number at the bottom of the clif
f.
d. Find the limit of the ratio ...at t approaches infinity. ...
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4.3.51 The population of lemmings L(t) at the top of a cliff is increasing with the given formula. However, lemmings leap off the cliff at a rate equal to 0.1L and pile up at the bottom.
a. Write a pure-time differential equation for the number of lemmings B(t) piled up at the bottom of the clif
f.
b. Solve using the initial condition B(0) = 0.
c. Graph the number of lemmings at the top and the number at the bottom of the clif
f.
d. Find the limit of the ratio ...at t approaches infinity. L ( t )=100t.
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4.3.52 The population of lemmings L(t) at the top of a cliff is increasing with the given formula. However, lemmings leap off the cliff at a rate equal to 0.1L and pile up at the bottom.
a. Write a pure-time differential equation for the number of lemmings B(t) piled up at the bottom of the clif
f.
b. Solve using the initial condition B(0) = 0.
c. Graph the number of lemmings at the top and the number at the bottom of the clif
f.
d. Find the limit of the ratio ...at t approaches infinity. ...
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4.3.53 Growth rates of insects depend on the temperature T . Suppose that the length of an insect L follows the differential equation ... with t measured
in days starting from January 1 and temperature measured in ?C. Insects
hatch with an initial size of 0.1 cm. For each of the following
equations for T (t),
a. Sketch a graph of the temperature over the course of a year.
b. Suppose an insect starts growing on January 1. How big will it be after 30 days?
c. Suppose an insect starts growing on June 1 (day 151). How big will it be after 30 days? T ( t ) = 0.001t (365 − t).
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4.3.54 Growth rates of insects depend on the temperature T . Suppose that the length of an insect L follows the differential equation ... with t measured
in days starting from January 1 and temperature measured in ?C. Insects
hatch with an initial size of 0.1 cm. For each of the following
equations for T (t),
a. Sketch a graph of the temperature over the course of a year.
b. Suppose an insect starts growing on January 1. How big will it be after 30 days?
c. Suppose an insect starts growing on June 1 (day 151). How big will it be after 30 days? ...
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4.3.55 Growth rates of insects depend on the temperature T . Suppose that the length of an insect L follows the differential equation ... with t measured
in days starting from January 1 and temperature measured in ?C. Insects
hatch with an initial size of 0.1 cm. For each of the following
equations for T (t),
a. Sketch a graph of the temperature over the course of a year.
b. Suppose an insect starts growing on January 1. How big will it be after 30 days?
c. Suppose an insect starts growing on June 1 (day 151). How big will it be after 30 days? ...
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4.3.56 Growth rates of insects depend on the temperature T . Suppose that the length of an insect L follows the differential equation ... with t measured
in days starting from January 1 and temperature measured in ?C. Insects
hatch with an initial size of 0.1 cm. For each of the following
equations for T (t),
a. Sketch a graph of the temperature over the course of a year.
b. Suppose an insect starts growing on January 1. How big will it be after 30 days?
c. Suppose an insect starts growing on June 1 (day 151). How big will it be after 30 days? ...
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4.3.57 Consider again the fish in Caribou Lake, growing according to ... where β =6.48 and α =0.09. However, suppose there is some variability among fish in the values of these two parameters.
a. Solve for growth trajectories of 5 fish with values of β evenly spread from 10% below to 10% above 6.48.
b. Solve for growth trajectories of 5 fish with values of α evenly spread from 10% below to 10% above 0.09.
c. Which parameter has a greater effect on the size of the fish?
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4.3.58
Clever genetic engineers design a set of fish that grow according to
the following equations. ... Suppose they all start at size 0.
Use your computer to sketch the growth of these fish for 0≤t ≤ 5. Then zoom in near t = 0. Could you tell which fish was which?
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