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Chapter 5

Integrals

Educators

+ 7 more educators

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 H.
Numerade Educator

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 H.
Numerade Educator

Problem 3

Verify by differentiation that the formula is correct.

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

Gregory H.
Numerade Educator

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 B.
Numerade Educator

Problem 5

Find the general indefinite integral.

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

Leon D.
Numerade Educator

Problem 6

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 7

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 8

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 9

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 10

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 11

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 12

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 13

Find the general indefinite integral.

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

Jacquelyn T.
Numerade Educator

Problem 14

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 15

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 16

Find the general indefinite integral.

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

Amrita B.
Numerade Educator

Problem 17

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

Problem 18

Find the general indefinite integral.

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

Gregory H.
Numerade Educator

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 H.
Numerade Educator

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 H.
Numerade Educator

Problem 21

Evaluate the integral.

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

Joseph R.
Numerade Educator

Problem 22

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 23

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 24

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 25

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 26

Evaluate the integral.

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

Ma. Theresa A.
Numerade Educator

Problem 27

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 28

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 29

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 30

Evaluate the integral.

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

Amrita B.
Numerade Educator

Problem 31

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 32

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 33

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 34

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 35

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 36

Evaluate the integral.

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

Joseph R.
Numerade Educator

Problem 37

Evaluate the integral.

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

Joseph R.
Numerade Educator

Problem 38

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 39

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 40

Evaluate the integral.

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

Amrita B.
Numerade Educator

Problem 41

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 42

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 43

Evaluate the integral.

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

Joseph R.
Numerade Educator

Problem 44

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 45

Evaluate the integral.

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

Gregory H.
Numerade Educator

Problem 46

Evaluate the integral.

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

Gregory H.
Numerade Educator

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 H.
Numerade Educator

Problem 48

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

Gregory H.
Numerade Educator

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 H.
Numerade Educator

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 R.
Numerade Educator

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 M.
Numerade Educator

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 B.
Numerade Educator

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 B.
Numerade Educator

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 H.
Numerade Educator

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 H.
Numerade Educator

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 B.
Numerade Educator

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 H.
Numerade Educator

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 H.
Numerade Educator

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 K.
Numerade Educator

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 A.
Numerade Educator

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 H.
Numerade Educator

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 H.
Numerade Educator

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 T.
Numerade Educator

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 R.
Numerade Educator

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 S.
Numerade Educator

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 B.
Numerade Educator

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 M.
Numerade Educator

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 B.
Numerade Educator

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 H.
Numerade Educator

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 M.
Numerade Educator

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 D.
Numerade Educator

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 B.
Numerade Educator

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 B.
Numerade Educator

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 L.
Numerade Educator