Calculus

Ron Larson, Robert P. Hostetler, Bruce H. Edwards

Chapter 4

Integration - all with Video Answers

Educators

Section 1

Antiderivatives and Indefinite Integration

Problem 1

Verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$
\int\left(-\frac{9}{x^{4}}\right) d x=\frac{3}{x^{3}}+C
$$

Mahendra Kumar

Mahendra Kumar

Numerade Educator

Problem 2

Verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$\int\left(4 x^{3}-\frac{1}{x^{2}}\right) d x=x^{4}+\frac{1}{x}+C$$

Mahendra Kumar

Numerade Educator

Problem 3

Verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$\int(x-2)(x+2) d x=\frac{1}{3} x^{3}-4 x+C$$

Mahendra Kumar

Numerade Educator

Problem 4

Verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$\int \frac{x^{2}-1}{x^{3 / 2}} d x=\frac{2\left(x^{2}+3\right)}{3 \sqrt{x}}+C$$

Mahendra Kumar

Numerade Educator

Problem 5

Find the general solution of the differential equation and check the result by differentiation.
$$\frac{d y}{d t}=3 t^{2}$$

Mahendra Kumar

Numerade Educator

Problem 6

Find the general solution of the differential equation and check the result by differentiation.
$$\frac{d r}{d \theta}=\pi$$

Mahendra Kumar

Numerade Educator

Problem 7

Find the general solution of the differential equation and check the result by differentiation.
$$\frac{d y}{d x}=x^{3 / 2}$$

Srikar Pasumarthy

Srikar Pasumarthy

Numerade Educator

Problem 8

Find the general solution of the differential equation and check the result by differentiation.
$$\frac{d y}{d x}=2 x^{-3}$$

Srikar Pasumarthy

Srikar Pasumarthy

Numerade Educator

Problem 9

Complete the table.
Original Integral
$$\int \sqrt[3]{x} d x$$

Mahendra Kumar

Numerade Educator

Problem 10

Complete the table.
Original Integral
$$\int \frac{1}{x^{2}} d x$$

Mahendra Kumar

Numerade Educator

Problem 11

Complete the table.
Original Integral
$$\int \frac{1}{x \sqrt{x}} d x$$

Mahendra Kumar

Numerade Educator

Problem 12

Complete the table.
Original Integral
$$\int x\left(x^{2}+3\right) d x$$

Mahendra Kumar

Numerade Educator

Problem 13

Complete the table.
Original Integral
$$\int \frac{1}{2 x^{3}} d x$$

Mahendra Kumar

Numerade Educator

Problem 14

Complete the table.
Original Integral
$$\int \frac{1}{(3 x)^{2}} d x$$

Mahendra Kumar

Numerade Educator

Problem 15

Find the indefinite integral and check the result by differentiation.
$$\int(x+3) d x$$

Mahendra Kumar

Numerade Educator

Problem 16

Find the indefinite integral and check the result by differentiation.
$\int(5-x) d x$$

Yaw Asomani

Numerade Educator

Problem 17

Find the indefinite integral and check the result by differentiation.
$$\int\left(2 x-3 x^{2}\right) d x$$

Mahendra Kumar

Numerade Educator

Problem 18

Find the indefinite integral and check the result by differentiation.
$$\int\left(4 x^{3}+6 x^{2}-1\right) d x$$

Mahendra Kumar

Numerade Educator

Problem 19

Find the indefinite integral and check the result by differentiation.
$$\int\left(x^{3}+2\right) d x$$

Yaw Asomani

Numerade Educator

Problem 20

Find the indefinite integral and check the result by differentiation.
$$\int\left(x^{3}-4 x+2\right) d x$$

Yaw Asomani

Numerade Educator

Problem 21

Find the indefinite integral and check the result by differentiation.
$$\int\left(x^{3 / 2}+2 x+1\right) d x$$

Srikar Pasumarthy

Srikar Pasumarthy

Numerade Educator

Problem 22

Find the indefinite integral and check the result by differentiation.
$$\int\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x$$

Yaw Asomani

Numerade Educator

Problem 23

Find the indefinite integral and check the result by differentiation.
$$\int \sqrt[3]{x^{2}} d x$$

Yaw Asomani

Numerade Educator

Problem 24

Find the indefinite integral and check the result by differentiation.
$$\left.\int(4]{x^{3}}+1\right) d x$$

Mahendra Kumar

Numerade Educator

Problem 25

Find the indefinite integral and check the result by differentiation.
$$\int \frac{1}{x^{3}} d x$$

Mahendra Kumar

Numerade Educator

Problem 26

Find the indefinite integral and check the result by differentiation.
$$\int \frac{1}{x^{4}} d x$$

Yaw Asomani

Numerade Educator

Problem 27

Find the indefinite integral and check the result by differentiation.
$$\int \frac{x^{2}+x+1}{\sqrt{x}} d x$$

Mahendra Kumar

Numerade Educator

Problem 28

Find the indefinite integral and check the result by differentiation.
$$\int \frac{x^{2}+2 x-3}{x^{4}} d x$$

Mahendra Kumar

Numerade Educator

Problem 29

Find the indefinite integral and check the result by differentiation.
$$\int(x+1)(3 x-2) d x$$

Srikar Pasumarthy

Srikar Pasumarthy

Numerade Educator

Problem 30

Find the indefinite integral and check the result by differentiation.
$$\int\left(2 t^{2}-1\right)^{2} d t$$

Mahendra Kumar

Numerade Educator

Problem 31

Find the indefinite integral and check the result by differentiation.
$$\int y^{2} \sqrt{y} d y$$

Mahendra Kumar

Numerade Educator

Problem 32

Find the indefinite integral and check the result by differentiation.
$$\int(1+3 t) t^{2} d t$$

Mahendra Kumar

Numerade Educator

Problem 33

Find the indefinite integral and check the result by differentiation.
$$\int d x$$

Mahendra Kumar

Numerade Educator

Problem 34

Find the indefinite integral and check the result by differentiation.
$$\int 3 d t$$

Mahendra Kumar

Numerade Educator

Problem 35

Find the indefinite integral and check the result by differentiation.
$$\int(2 \sin x+3 \cos x) d x$$

Mahendra Kumar

Numerade Educator

Problem 36

Find the indefinite integral and check the result by differentiation.
$$\int\left(t^{2}-\sin t\right) d t$$

Mahendra Kumar

Numerade Educator

Problem 37

Find the indefinite integral and check the result by differentiation.
$$\int(1-\csc t \cot t) d t$$

Mahendra Kumar

Numerade Educator

Problem 38

Find the indefinite integral and check the result by differentiation.
$$\int\left(\theta^{2}+\sec ^{2} \theta\right) d \theta$$

Srikar Pasumarthy

Srikar Pasumarthy

Numerade Educator

Problem 39

Find the indefinite integral and check the result by differentiation.
$$\int\left(\sec ^{2} \theta-\sin \theta\right) d \theta$$

Mahendra Kumar

Numerade Educator

Problem 40

Find the indefinite integral and check the result by differentiation.
$$\int \sec y(\tan y-\sec y) d y$$

Srikar Pasumarthy

Srikar Pasumarthy

Numerade Educator

Problem 41

Find the indefinite integral and check the result by differentiation.
$$\int\left(\tan ^{2} y+1\right) d y$$

Srikar Pasumarthy

Srikar Pasumarthy

Numerade Educator

Problem 42

Find the indefinite integral and check the result by differentiation.
$$\int \frac{\cos x}{1-\cos ^{2} x} d x$$

Mahendra Kumar

Numerade Educator

Problem 43

The graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, select the MathGraph button.

Mahendra Kumar

Numerade Educator

Problem 44

The graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, select the MathGraph button.

Stanley Enemuo

Numerade Educator

Problem 45

The graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, select the MathGraph button.

Mahendra Kumar

Numerade Educator

Problem 46

The graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, select the MathGraph button.

Mahendra Kumar

Numerade Educator

Problem 47

Find the equation for $y$, given the derivative and the indicated point on the curve.

Mahendra Kumar

Numerade Educator

Problem 48

Find the equation for $y$, given the derivative and the indicated point on the curve.

Mahendra Kumar

Numerade Educator

Problem 49

A differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, select the MathGraph button.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$
\frac{d y}{d x}=\frac{1}{2} x-1, \quad(4,2)
$$

Stanley Enemuo

Numerade Educator

Problem 50

A differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, select the MathGraph button.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$
\frac{d y}{d x}=x^{2}-1, \quad(-1,3)
$$

Stanley Enemuo

Numerade Educator

Problem 51

A differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, select the MathGraph button.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$
\frac{d y}{d x}=\cos x,(0,4)
$$

Stanley Enemuo

Numerade Educator

Problem 52

A differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, select the MathGraph button.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$
\frac{d y}{d x}=-\frac{1}{x^{2}}, x>0,(1,3)
$$

Stanley Enemuo

Numerade Educator

Problem 53

Slope Fields (a) use a graphing utility to graph a slope field for the differential equation, (b) use integration and the given point to find the particular solution of the differential equation, and (c) graph the solution and the slope field in the same viewing window.
$$\frac{d y}{d x}=2 x,(-2,-2)$$

Stanley Enemuo

Numerade Educator

Problem 54

Slope Fields (a) use a graphing utility to graph a slope field for the differential equation, (b) use integration and the given point to find the particular solution of the differential equation, and (c) graph the solution and the slope field in the same viewing window.
$$\frac{d y}{d x}=2 \sqrt{x},(4,12)$$

Stanley Enemuo

Numerade Educator

Problem 55

Solve the differential equation.
$$f^{\prime}(x)=4 x, f(0)=6$$

Mahendra Kumar

Numerade Educator

Problem 56

Solve the differential equation.
$$g^{\prime}(x)=6 x^{2}, g(0)=-1$$

Mahendra Kumar

Numerade Educator

Problem 57

Solve the differential equation.
$$h^{\prime}(t)=8 t^{3}+5, h(1)=-4$$

Mahendra Kumar

Numerade Educator

Problem 58

Solve the differential equation.
$$f^{\prime}(s)=6 s-8 s^{3}, f(2)=3$$

Mahendra Kumar

Numerade Educator

Problem 59

Solve the differential equation.
$$f^{\prime \prime}(x)=2, f^{\prime}(2)=5, f(2)=10$$

Mahendra Kumar

Numerade Educator

Problem 60

Solve the differential equation.
$$f^{\prime \prime}(x)=x^{2}, f^{\prime}(0)=6, f(0)=3$$

Mahendra Kumar

Numerade Educator

Problem 61

Solve the differential equation.
$$f^{\prime \prime}(x)=x^{-3 / 2}, f^{\prime}(4)=2, f(0)=0$$

Mahendra Kumar

Numerade Educator

Problem 62

Solve the differential equation.
$$f^{N}(x)=\sin x, f^{\prime}(0)=1, f(0)=6$$

Mahendra Kumar

Numerade Educator

Problem 63

An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by $d h / d t=1.5 t+5$ where $t$ is the time in years and $h$ is the height in centimeters. The seedlings are 12 centimeters tall when planted $(t=0)$.
(a) Find the height after $t$ years.
(b) How tall are the shrubs when they are sold?

Danielle Fairburn

Danielle Fairburn

Numerade Educator

Problem 64

The rate of growth $d P / d t$ of a population of bacteria is proportional to the square root of $t$, where $P$ is the population size and $t$ is the time in days $(0 \leq t \leq 10)$. That is, $d P / d t=k \sqrt{t}$. The initial size of the population is $500 .$ After 1 day the population has grown to 600 . Estimate the population after 7 days.

Allan Hungria

Numerade Educator

Problem 65

Use the graph of $f^{\prime}$ shown in the figure to answer the following, given that $f(0)=-4$.
(a) Approximate the slope of $f$ at $x=4$. Explain.
(b) Is it possible that $f(2)=-1 ?$ Explain.
(c) Is $f(5)-f(4)>0 ?$ Explain.
(d) Approximate the value of $x$ where $f$ is maximum. Explain.
(e) Approximate any intervals in which the graph of $f$ is concave upward and any intervals in which it is concave downward. Approximate the $x$ -coordinates of any points of inflection.
(f) Approximate the $x$ -coordinate of the minimum of $f^{\prime \prime}(x)$.
(g) Sketch an approximate graph of $f .$ To print an enlarged copy of the graph, select the MathGraph button.

Stanley Enemuo

Numerade Educator

Problem 66

The graphs of $f$ and $f^{\prime}$ each pass through the origin. Use the graph of $f^{\prime \prime}$ shown in the figure to sketch the graphs of $f$ and $f^{\prime} .$ To print an enlarged copy of the graph, select the MathGraph button.

Stanley Enemuo

Numerade Educator

Problem 67

A ball is thrown vertically upward from a height of 6 feet with an initial velocity of 60 feet per second. How high will the ball go?

William Semus

Numerade Educator

Problem 68

Show that the height above the ground of an object thrown upward from a point $s_{0}$ feet above the ground with an initial velocity of $v_{0}$ feet per second is given by the function
$f(t)=-16 t^{2}+v_{0} t+s_{0}$

Stanley Enemuo

Numerade Educator

Problem 69

With what initial velocity must an object be thrown upward (from ground level) to reach the top of the Washington Monument (approximately 550 feet)?

Khushbu Rani

Numerade Educator

Problem 70

A balloon, rising vertically with a velocity of 16 feet per second, releases a sandbag at the instant it is 64 feet above the ground.
(a) How many seconds after its release will the bag strike the ground?
(b) At what velocity will it hit the ground?

Sherrie Fenner

Numerade Educator

Problem 71

Show that the height above the ground of an object thrown upward from a point $s_{0}$ meters above the ground with an initial velocity of $v_{0}$ meters per second is given by the function
$f(t)=-4.9 t^{2}+v_{0} t+s_{0}$

Stanley Enemuo

Numerade Educator

Problem 72

The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of the time $t$ in seconds. How long will it take the rock to hit the canyon floor?

Isabella Cooper

Isabella Cooper

Numerade Educator

Problem 73

A baseball is thrown upward from a height of 2 meters with an initial velocity of 10 meters per second. Determine its maximum height.

Mahendra Kumar

Numerade Educator

Problem 74

With what initial velocity must an object be thrown upward (from a height of 2 meters) to reach a maximum height of 200 meters?

Sherrie Fenner

Numerade Educator

Problem 75

Lunar Gravity On the moon, the acceleration due to gravity is $-1.6$ meters per second per second. A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?

Gregory Higby

Numerade Educator

Problem 76

The minimum velocity required for an object to escape Earth's gravitational pull is obtained from the solution of the equation $\int v d v=-G M \int \frac{1}{y^{2}} d y$
where $v$ is the velocity of the object projected from Earth, $y$ is the distance from the center of Earth, $G$ is the gravitational constant, and $M$ is the mass of Earth. Show that $v$ and $y$ are related by the equation
$v^{2}=v_{0}^{2}+2 G M\left(\frac{1}{y}-\frac{1}{R}\right)$
where $v_{0}$ is the initial velocity of the object and $R$ is the radius of Earth.

Allan Hungria

Numerade Educator

Problem 77

$x(t)=t^{3}-6 t^{2}+9 t-2, \quad 0 \leq t \leq 5$
(a) Find the velocity and acceleration of the particle.
(b) Find the open $t$ -intervals on which the particle is moving to the right.
(c) Find the velocity of the particle when the acceleration is $0 .$

Mahendra Kumar

Numerade Educator

Problem 78

Repeat Exercise 77 for the position function
$x(t)=(t-1)(t-3)^{2}, \quad 0 \leq t \leq 5$

Mahendra Kumar

Numerade Educator

Problem 79

A particle moves along the $x$ -axis at a velocity of $v(t)=1 / \sqrt{t}$ $t>0$. At time $t=1$, its position is $x=4$. Find the acceleration and position functions for the particle.

Sherrie Fenner

Numerade Educator

Problem 80

A particle, initially at rest, moves along the $x$ -axis such that its acceleration at time $t>0$ is given by $a(t)=\cos t .$ At the time $t=0$, its position is $x=3$.
(a) Find the velocity and position functions for the particle.
(b) Find the values of $t$ for which the particle is at rest.

William Semus

Numerade Educator

Problem 81

The maker of an automobile advertises that it takes 13 seconds to accelerate from 25 kilometers per hour to 80 kilometers per hour. Assuming constant acceleration, compute the following.
(a) The acceleration in meters per second per second
(b) The distance the car travels during the 13 seconds

Stanley Enemuo

Numerade Educator

Problem 82

A car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied.
(a) How far has the car moved when its speed has been reduced to 30 miles per hour?
(b) How far has the car moved when its speed has been reduced to 15 miles per hour?
(c) Draw the real number line from 0 to 132 , and plot the points found in parts (a) and (b). What can you conclude?

Allan Hungria

Numerade Educator

Problem 83

At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car.
(a) How far beyond its starting point will the car pass the truck?
(b) How fast will the car be traveling when it passes the truck?

Sherrie Fenner

Numerade Educator

Problem 84

The table shows the velocities (in miles per hour) of two cars on an entrance ramp to an interstate highway. The time $t$ is in seconds.
$$
\begin{array}{|l|c|c|c|c|c|c|c|}
\hline t & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline v_{1} & 0 & 2.5 & 7 & 16 & 29 & 45 & 65 \\
\hline v_{2} & 0 & 21 & 38 & 51 & 60 & 64 & 65 \\
\hline
\end{array}
$$
(a) Rewrite the table converting miles per hour to feet per second.
(b) Use the regression capabilities of a graphing utility to find quadratic models for the data in part (a).
(c) Approximate the distance traveled by each car during the 30 seconds. Explain the difference in the distances.

Stanley Enemuo

Numerade Educator

Problem 85

Assume that a fully loaded plane starting from rest has a constant acceleration while moving down a runway. The plane requires $0.7$ mile of runway and a speed of 160 miles per hour in order to lift off. What is the plane's acceleration?

Allan Hungria

Numerade Educator

Problem 86

Two airplanes are in a straight-line landing pattern and, according to FAA regulations, must keep at least a three-mile separation. Airplane $A$ is 10 miles from touchdown and is gradually decreasing its speed from 150 miles per hour to a landing speed of 100 miles per hour. Airplane $\mathrm{B}$ is 17 miles from touchdown and is gradually decreasing its speed from 250 miles per hour to a landing speed of 115 miles per hour.
(a) Assuming the deceleration of each airplane is constant, find the position functions $s_{1}$ and $s_{2}$ for airplane $A$ and airplane
B. Let $t=0$ represent the times when the airplanes are 10 and 17 miles from the airport.
(b) Use a graphing utility to graph the position functions.
(c) Find a formula for the magnitude of the distance $d$ between the two airplanes as a function of $t .$ Use a graphing utility to graph $d .$ Is $d<3$ for some time prior to the landing of airplane $A$ ? If so, find that time.

Stanley Enemuo

Numerade Educator

Problem 87

Each antiderivative of an $n$ th-degree polynomial function is an $(n+1)$ th-degree polynomial function.

Mahendra Kumar

Numerade Educator

Problem 88

If $p(x)$ is a polynomial function, then $p$ has exactly one antiderivative whose graph contains the origin.

Mahendra Kumar

Numerade Educator

Problem 89

If $F(x)$ and $G(x)$ are antiderivatives of $f(x)$, then $F(x)=G(x)+C$

Mahendra Kumar

Numerade Educator

Problem 90

If $f^{\prime}(x)=g(x)$, then $\int g(x) d x=f(x)+C$.

Mahendra Kumar

Numerade Educator

Problem 91

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$$\int f(x) g(x) d x=\int f(x) d x \int g(x) d x$$

Mahendra Kumar

Numerade Educator

Problem 92

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The antiderivative of $f(x)$ is unique.

Isabella Cooper

Isabella Cooper

Numerade Educator

Problem 93

Find a function $f$ such that the graph of $f$ has a horizontal tangent at $(2,0)$ and $f^{\prime \prime}(x)=2 x$.

Sherrie Fenner

Numerade Educator

Problem 94

The graph of $f^{\prime}$ is shown. Sketch the graph of $f$ given that $f$ is continuous and $f(0)=1$.

Gregory Higby

Numerade Educator

Problem 95

If $f^{\prime}(x)=\left\{\begin{array}{cc}1, & 0 \leq x<2 \\ 3 x, & 2 \leq x \leq 5\end{array}, f\right.$ is continuous, and $f(1)=3$, find $f .$ Is $f$ differentiable at $x=2 ?$

Gregory Higby

Numerade Educator

Problem 96

Let $s(x)$ and $c(x)$ be two functions satisfying $s^{\prime}(x)=c(x)$ and $c^{\prime}(x)=-s(x)$ for all $x .$ If $s(0)=0$ and $c(0)=1$, prove that $[s(x)]^{2}+[c(x)]^{2}=1$.

Carson Merrill

Numerade Educator

Problem 97

Suppose $f$ and $g$ are nonconstant, differentiable, real-valued functions on $R .$ Furthermore, suppose that for each pair of real numbers $x$ and $y, f(x+y)=f(x) f(y)-g(x) g(y)$ and $g(x+y)=f(x) g(y)+g(x) f(y) .$ If $f^{\prime}(0)=0$, prove that $(f(x))^{2}+(g(x))^{2}=1$ for all $x$.

Stanley Enemuo

Numerade Educator