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  • Calculus: Early Transcendentals
  • Integrals

Calculus: Early Transcendentals

James Stewart

Chapter 5

Integrals - all with Video Answers

Educators

+ 8 more educators

Section 4

Indefinite Integrals and the Net Change Theorem

02:38

Problem 1

Verify by differentiation that the formula is correct.

$ \displaystyle \int \frac{1}{x^2 \sqrt{1 + x^2}} \,dx = - \frac{\sqrt{1 + x^2}}{x} + C $

Gregory Higby
Gregory Higby
Numerade Educator
02:31

Problem 2

Verify by differentiation that the formula is correct.

$ \displaystyle \int \cos^2 x \,dx = \frac{1}{2}x + \frac{1}{4}\sin 2x+ C $

Gregory Higby
Gregory Higby
Numerade Educator
01:26

Problem 3

Verify by differentiation that the formula is correct.

$ \displaystyle \int \tan^2 x \,dx = \tan x - x + C $

Gregory Higby
Gregory Higby
Numerade Educator
01:02

Problem 4

Verify by differentiation that the formula is correct.

$ \displaystyle \int x\sqrt{a + bx} \,dx = \frac{2}{15b^2}(3bx - 2a)(a + bx)^{3/2} + C $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
04:19

Problem 5

Find the general indefinite integral.

$ \displaystyle \int (x^{1.3} + 7x^{2.5}) \, dx $

Leon Druch
Leon Druch
Numerade Educator
01:40

Problem 6

Find the general indefinite integral.

$ \displaystyle \int \sqrt[4]{x^5} \, dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:38

Problem 7

Find the general indefinite integral.

$ \displaystyle \int (5 + \frac{2}{3}x^2 + \frac{3}{4}x^3) \, dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:30

Problem 8

Find the general indefinite integral.

$ \displaystyle \int (u^6 - 2u^5 - u^3 + \frac{2}{7}) \, du $

Gregory Higby
Gregory Higby
Numerade Educator
01:43

Problem 9

Find the general indefinite integral.

$ \displaystyle \int (u + 4)(2u + 1) \, du $

Gregory Higby
Gregory Higby
Numerade Educator
02:09

Problem 10

Find the general indefinite integral.

$ \displaystyle \int \sqrt{t} (t^2 + 3t + 2) \, dt $

Gregory Higby
Gregory Higby
Numerade Educator
01:37

Problem 11

Find the general indefinite integral.

$ \displaystyle \int \frac{1 + \sqrt{x} + x}{x} \, dx $

Mary Wakumoto
Mary Wakumoto
Numerade Educator
01:37

Problem 12

Find the general indefinite integral.

$ \displaystyle \int \biggl( x^2 + 1 + \frac{1}{x^2 + 1} \biggr)\, dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:59

Problem 13

Find the general indefinite integral.

$ \displaystyle \int (\sin x + \sinh x)\, dx $

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
02:03

Problem 14

Find the general indefinite integral.

$ \displaystyle \int \biggl( \frac{1 + r}{r} \biggr)^2 \, dr $

Gregory Higby
Gregory Higby
Numerade Educator
01:32

Problem 15

Find the general indefinite integral.

$ \displaystyle \int (2 + \tan^2 \theta)\, d\theta $

Gregory Higby
Gregory Higby
Numerade Educator
00:46

Problem 16

Find the general indefinite integral.

$ \displaystyle \int \sec t (\sec t + \tan t)\, dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:35

Problem 17

Find the general indefinite integral.

$ \displaystyle \int 2^t (1 + 5^t)\, dt $

Gregory Higby
Gregory Higby
Numerade Educator
01:15

Problem 18

Find the general indefinite integral.

$ \displaystyle \int \frac{\sin 2x}{\sin x}\, dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:09

Problem 19

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen.

$ \displaystyle \int \biggl( \cos x + \frac{1}{2}x \biggr) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:19

Problem 20

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen.

$ \displaystyle \int (e^x - 2x^2) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
02:12

Problem 21

Evaluate the integral.

$ \displaystyle \int^3_{-2} (x^2 - 3) \,dx $

Joseph Russell
Joseph Russell
Numerade Educator
01:48

Problem 22

Evaluate the integral.

$ \displaystyle \int^2_{1} (4x^3 - 3x^2 + 2x) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
02:39

Problem 23

Evaluate the integral.

$ \displaystyle \int^0_{-2} \biggl( \frac{1}{2}t^4 + \frac{1}{4}t^3 - t \biggr) \,dt $

Gregory Higby
Gregory Higby
Numerade Educator
02:00

Problem 24

Evaluate the integral.

$ \displaystyle \int^{3}_{0} (1 + 6w^2 - 10w^4) \,dw $

Gregory Higby
Gregory Higby
Numerade Educator
02:14

Problem 25

Evaluate the integral.

$ \displaystyle \int^{2}_{0} (2x - 3)(4x^2 + 1) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
View

Problem 26

Evaluate the integral.

$ \displaystyle \int^{1}_{-1} t(1 - t)^2 \,dt $

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:28

Problem 27

Evaluate the integral.

$ \displaystyle \int^{\pi}_{0} (5e^x + 3\sin x) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:43

Problem 28

Evaluate the integral.

$ \displaystyle \int^{2}_{1} \biggl( \frac{1}{x^2} - \frac{4}{x^3} \biggr) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
02:06

Problem 29

Evaluate the integral.

$ \displaystyle \int^{4}_{1} \biggl( \frac{4 + 6u}{\sqrt{u}} \biggr) \,du $

Gregory Higby
Gregory Higby
Numerade Educator
00:44

Problem 30

Evaluate the integral.

$ \displaystyle \int^{1}_{0} \frac{4}{1 + p^2} \,dp $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:09

Problem 31

Evaluate the integral.

$ \displaystyle \int^{1}_{0} x \bigl( \sqrt[3]{x} + \sqrt[4]{x} \bigr) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
02:50

Problem 32

Evaluate the integral.

$ \displaystyle \int^{4}_{1} \frac{\sqrt{y} - y}{y^2} \,dy $

Gregory Higby
Gregory Higby
Numerade Educator
01:43

Problem 33

Evaluate the integral.

$ \displaystyle \int^{2}_{1} \biggl( \frac{x}{2} - \frac{2}{x}\biggr) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
02:05

Problem 34

Evaluate the integral.

$ \displaystyle \int^{1}_{0} (5x - 5^x) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:51

Problem 35

Evaluate the integral.

$ \displaystyle \int^{1}_{0} (x^{10} + 10^x)\,dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:30

Problem 36

Evaluate the integral.

$ \displaystyle \int^{\pi/4}_{0} \sec \theta \tan \theta \,d\theta $

Joseph Russell
Joseph Russell
Numerade Educator
02:21

Problem 37

Evaluate the integral.

$ \displaystyle \int^{\pi/4}_{0} \frac{1 + \cos^2 \theta}{\cos^2 \theta} \,d\theta $

Joseph Russell
Joseph Russell
Numerade Educator
02:22

Problem 38

Evaluate the integral.

$ \displaystyle \int^{\pi/3}_{0} \frac{\sin \theta + \sin \theta \tan^2 \theta}{\sec^2 \theta} \,d\theta $

Gregory Higby
Gregory Higby
Numerade Educator
02:57

Problem 39

Evaluate the integral.

$ \displaystyle \int^{8}_{1} \frac{2 + t}{\sqrt[3]{t^2}} \,dt $

Gregory Higby
Gregory Higby
Numerade Educator
00:40

Problem 40

Evaluate the integral.

$ \displaystyle \int^{10}_{-10} \frac{2e^x}{\sinh x + \cosh x} \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:43

Problem 41

Evaluate the integral.

$ \displaystyle \int^{\sqrt{3}/2}_{0} \frac{dr}{\sqrt{1 - r^2}} $

Gregory Higby
Gregory Higby
Numerade Educator
02:53

Problem 42

Evaluate the integral.

$ \displaystyle \int^{2}_{1} \frac{(x - 1)^3}{x^2} \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:52

Problem 43

Evaluate the integral.

$ \displaystyle \int^{1/\sqrt{3}}_{0} \frac{t^2 - 1}{t^4 - 1} \,dt $

Joseph Russell
Joseph Russell
Numerade Educator
01:46

Problem 44

Evaluate the integral.

$ \displaystyle \int^{2}_{0} \mid 2x - 1 \mid \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
02:10

Problem 45

Evaluate the integral.

$ \displaystyle \int^{2}_{-1} \bigl( x - 2 \mid x \mid \bigr) \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
02:07

Problem 46

Evaluate the integral.

$ \displaystyle \int^{3\pi/2}_{0} \mid \sin x \mid \,dx $

Gregory Higby
Gregory Higby
Numerade Educator
01:58

Problem 47

Use a graph to estimate the $ x $-intercepts of the curve $ y = 1 - 2x - 5x^4 $. Then use this information to estimate the area of the region that lies under the curve and above the $ x $-axis.

Gregory Higby
Gregory Higby
Numerade Educator
02:09

Problem 48

Repeat Exercise 47 for the curve $ y = (x^2 + 1)^{-1} - x^4 $.

Gregory Higby
Gregory Higby
Numerade Educator
01:34

Problem 49

The area of the region that lies to the right of the $ y $-axis and to the left of the parabola $ x = 2y - y^2 $ (the shaded region in the figure) is given by the integral $ \displaystyle \int^2_0 (2y - y^2) \, dy $. (Turn your head clockwise and think of the region as lying below the curve $ x = 2y - y^2 $ from $ y = 0 $ to $ y = 2 $.) Find the area of the region.

Anthony Han
Anthony Han
Numerade Educator
02:18

Problem 50

The boundaries of the shaded region are the $ y $-axis, the line $ y = 1 $, and the curve $ y = \sqrt[4]{x} $. Find the area of this region by writing $ x $ as a function of $ y $ and integrating with respect to $ y $ (as in Exercise 49).

Joseph Russell
Joseph Russell
Numerade Educator
01:27

Problem 51

If $ w'(t) $ is the rate of growth of a child in pounds per year, what does$ \displaystyle \int^{10}_5 w'(t) \,dt $ represent?

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
00:39

Problem 52

The current in a wire is defined as the derivative of the charge: $ I(t) = Q'(t) $. (See Example 3.7.3.) What does $ \displaystyle \int^b_a I(t) \, dt $ represent?

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:43

Problem 53

If oil leaks from a tank at a rate of $ r(t) $ gallons per minute at time $ t $, what does $ \displaystyle \int^{120}_0 r(t) \, dt $ represent?

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:18

Problem 54

A honeybee population starts with 100 bees and increases at a rate of $ n'(t) $ bees per week. What does $ \displaystyle 100 + \int^{15}_0 n'(t) \, dt $ represent?

Gregory Higby
Gregory Higby
Numerade Educator
01:24

Problem 55

In Section 4.7 we defined the marginal revenue function $ R'(x) $ as the derivative of the revenue function $ R(x) $, where $ x $ is the number of units sold. What does $ \displaystyle \int^{5000}_{1000} R'(x) \, dx $ represent?

Gregory Higby
Gregory Higby
Numerade Educator
00:28

Problem 56

If $ f(x) $ is the slope of a trail at a distance of $ x $ miles from the start of the trail, what does $ \displaystyle \int^5_3 f(x) \, dx $ represent?

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:21

Problem 57

If $ x $ is measured in meters and $ f(x) $ is measured in newtons, what are the units for $ \displaystyle \int^{100}_0 f(x) \, dx $?

Gregory Higby
Gregory Higby
Numerade Educator
02:02

Problem 58

If the units for $ x $ are feet and the units for $ a(x) $ are pounds per foot, what are the units for $ da/dx $? What units does $ \displaystyle \int^8_2 a(x) \, dx $ have?

Linda Hand
Linda Hand
Numerade Educator
11:14

Problem 59

The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval.

$ v(t) = 3t - 5 $, $ 0 \le t \le 3 $

Jen Kelly
Jen Kelly
Numerade Educator
View

Problem 60

The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval.

$ v(t) = t^2 - 2t - 3 $, $ 2 \le t \le 4 $

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
05:19

Problem 61

The acceleration function (in $ m/s^2 $) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time $ t $ and (b) the distance traveled during the given time interval.

$ a(t) = t + 4 $, $ v(0) = 5 $, $ 0 \le t \le 10 $

Dakarai Holcomb
Dakarai Holcomb
Numerade Educator
05:19

Problem 62

The acceleration function (in $ m/s^2 $) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time $ t $ and (b) the distance traveled during the given time interval.

$ a(t) = 2t + 3 $, $ v(0) = -4 $, $ 0 \le t \le 3 $

Gregory Higby
Gregory Higby
Numerade Educator
03:10

Problem 63

The linear density of a rod of length $ 4 m $ is given by $ \rho (x) = 9 + 2 \sqrt{x} $ measured in kilograms per meter, where $ x $ is measured in meters from one end of the rod. Find the total mass of the rod.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:43

Problem 64

Water flows from the bottom of a storage tank at a rate of $ r(t) = 200 - 4t $ liters per minute, where $ 0 \le t \le 50 $. Find the amount of water that flows from the tank during the first 10 minutes.

Joseph Russell
Joseph Russell
Numerade Educator
05:30

Problem 65

The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car.

Aparna Shakti
Aparna Shakti
Numerade Educator
08:30

Problem 66

Suppose that a volcano is erupting and readings of the rate $ r(t) $ at which solid materials are spewed into the atmosphere are given in the table. The time $ t $ is measured in seconds and the units for $ r(t) $ are tonnes (metric tons) per second.

(a) Give upper and lower estimates for the total quantity $ Q(6) $ of erupted materials alter six seconds.
(b) Use the Midpoint Rule to estimate $ Q(6) $.

Carolyn Behr-Jerome
Carolyn Behr-Jerome
Numerade Educator
05:48

Problem 67

The marginal cost of manufacturing $ x $ yards of a certain fabric is
$$ C'(x) = 3 - 0.01x + 0.000006x^2 $$
(in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards.

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
04:08

Problem 68

Water flows into and out of a storage tank. A graph of the rate of change $ r(t) $ of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time $ t = 0 $ is $ 25,000 L $, use the Midpoint Rule to estimate the amount of water in the tank four days later.

Carolyn Behr-Jerome
Carolyn Behr-Jerome
Numerade Educator
03:12

Problem 69

The graph of the acceleration $ a(t) $ of a car measured in $ ft/s^2 $ is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the six-second time interval.

Yuki Hotta
Yuki Hotta
Numerade Educator
07:22

Problem 70

Lake Lanier in Georgia, USA, is a reservoir created by Buford Dam on the Chattahoochee River. The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 AM by the US Army Corps of Engineers. Use the Midpoint Rule to estimate the amount of water that flowed into Lake Lanier from July 18th, 2013, at 7:30 AM to July 26th at 7:30 AM.

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
03:07

Problem 71

A bacteria population is 4000 at time $ t = 0 $ and its rate of growth is $ 1000 \cdot 2^t $ bacteria per hour after $ t $ hours. What is the population after one hour?

Leon Druch
Leon Druch
Numerade Educator
01:24

Problem 72

Shown is the graph of traffic on an Internet service provider's T1 data line from midnight to 8:00 AM. $ D $ is the data throughput, measured in megabits per second. Use the Midpoint Rule to estimate the total amount of data transmitted during that time period.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:25

Problem 73

Shown is the power consumption in the province of Ontario, Canada, for December 9, 2004 ($ P $ is measured in megawatts; $ t $ is measured in hours starting at midnight). Using the fact that power is the rate of change of energy, estimate the energy used on that day.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:03

Problem 74

On May 7, 1992, the space shuttle $ Endeavour $ was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rock boosters.

(a) Use a graphing calculator or computer to model these data by a third-degree polynomial.
(b) Use the model in part (a) to estimate the height reached by the $ Endeavour $, 125 seconds after liftoff.

Frank Lin
Frank Lin
Numerade Educator

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