ApplicationsThe surface generated by an astroid The graph

Problem 116PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 1PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 1AAE Evaluating Integrals Evaluate the integrals.

Problem 1QGY What is the formula for integration by parts? Where does it come from? Why might you want to use it?

Problem 2AAE Evaluating Integrals Evaluate the integrals.

Problem 2PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 2QGY When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ?f(x)dx?

Problem 3AAE Evaluating Integrals Evaluate the integrals.

Problem 3PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 4AAE Evaluating Integrals Evaluate the integrals.

Problem 4PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 3QGY If an integrand is a product of the form sinn x cosm x, where m and n are nonnegative integers, how do you evaluate the integral? Give a specific example of each case.

Problem 5AAE Evaluating Integrals Evaluate the integrals.

Problem 4QGY What substitutions are made to evaluate integrals of sin mx sin nx, sin mx cos nx, and cos mx cos nx? Give an example of each case.

Problem 5QGY What substitutions are sometimes used to transform integrals involving into integrals that can be evaluated directly? Give an example of each case.

Problem 6AAE Evaluating Integrals Evaluate the integrals.

Problem 6PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 5PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 6QGY What restrictions can you place on the variables involved in the three basic trigonometric substitutions to make sure the substitutions are reversible (have inverses)?

Problem 7AAE Evaluating Integrals Evaluate the limits

Problem 7PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 8AAE Evaluating Integrals Evaluate the limits

Problem 7QGY What is the goal of the method of partial fractions?

Problem 8QGY When the degree of a polynomial ƒ(x) is less than the degree of a polynomial g(x), how do you write ƒ(x) / g(x) as a sum of partial fractions if g(x) a. is a product of distinct linear factors? b. consists of a repeated linear factor? c. contains an irreducible quadratic factor? What do you do if the degree of ƒ is not less than the degree of g?

Problem 8PE Integration by Parts Evaluate the integrals using integration by parts.

Problem 9AAE Evaluating Integrals Evaluate the limits by identifying them with definite integrals and evaluating the integrals.

Problem 9QGY How are integral tables typically used? What do you do if a particular integral you want to evaluate is not listed in the table?

Problem 10PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 10AAE Evaluating Integrals Evaluate the limits by identifying them with definite integrals and evaluating the integrals.

Problem 9PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 10QGY What is a reduction formula? How are reduction formulas used? Give an example.

Problem 11AAE Applications Finding arc length Find the length of the curve

Problem 11PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 12AAE Applications Finding arc length Find the length of the graph of the function

Problem 11QGY How would you compare the relative merits of Simpson’s Rule and the Trapezoidal Rule?

Problem 12PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 12QGY What is an improper integral of Type I? Type II? How are the values of various types of improper integrals defined? Give examples.

Problem 13AAE Applications Finding volume The region in the first quadrant that is enclosed by the x-axis and the curve is revolved about the y-axis to generate a solid. Find the volume of the solid.

Problem 13PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 14AAE Applications Finding volume The region in the first quadrant that is enclosed by the x-axis, the curve and the lines x = 1 and x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid.

Problem 13QGY What tests are available for determining the convergence and divergence of improper integrals that cannot be evaluated directly? Give examples of their use.

Problem 14PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 14QGY What is a random variable? What is a continuous random variable? Give some specific examples.

Problem 15AAE Applications Finding volume The region in the first quadrant enclosed by the coordinate axes, the curve y = ex and x = 1 the line is revolved about the y-axis to generate a solid. Find the volume of the solid.

Problem 15PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 15QGY What is a probability density function? What is the probability that a continuous random variable has a value in the interval [ c, d ]?

Problem 16AAE Applications Finding volume The region in the first quadrant that is bounded above by the curve y = ex - 1, below by the x-axis, and on the right by the line x = ln 2 is revolved about the line x = ln 2 to generate a solid. Find the volume of the solid.

Problem 16PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 16QGY What is an exponentially decreasing probability density function? What are some typical events that might be modeled by this distribution? What do we mean when we say such distributions are memoryless?

Problem 17PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 17AAE Applications Finding volume Let R be the “triangular” region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.Find the volume of the solid generated by revolving R about a. the x-axis. b. the line y = 1

Problem 18AAE Applications Finding volume (Continuation of Exercise 17. ) Find the volume of the solid generated by revolving the region R about a. the y-axis. b. the line x = 1 Reference: Finding volume Let R be the “triangular” region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.Find the volume of the solid generated by revolving R about a. the x-axis. b. the line y = 1

Problem 17QGY What is the expected value of a continuous random variable? What is the expected value of an exponentially distributed random variable?

Problem 18PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 18QGY What is the median of a continuous random variable? What is the median of an exponential distribution?

Problem 19AAE Applications Finding volume The region between the x-axis and the curve is revolved about the x-axis to generate the solid shown here. a. Show that ƒ is continuous at x = 0. b. Find the volume of the solid.

Problem 19PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 19QGY What does the variance of a random variable measure? What is the standard deviation of a continuous random variable X?

Problem 20PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 20QGY What probability density function describes the normal distribution? What are some examples typically modeled by a normal distribution? How do we usually calculate probabilities for a normal distribution?

Problem 21AAE Applications Centroid of a region Find the centroid of the region in the first quadrant that is bounded below by the x-axis, above by the curve y = ln x, and on the right by the line x = e.

Problem 20AAE Applications Finding volume The infinite region bounded by the coordinate axes and the curve y = -ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

Problem 21QGY In a normal distribution, what percentage of the population lies within 1 standard deviation of the mean? Within 2 standard deviations?

Problem 21PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 22AAE Applications Centroid of a region Find the centroid of the region in the plane enclosed by the curves and the lines x = 0 and x = 1.

Problem 22PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 23AAE Applications Length of a curve Find the length of the curve y = ln x from x = 1 to x = e.

Problem 23PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 24PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 25AAE Applications The surface generated by an astroid The graph of the equation x2/3 + y2/3 = 1 is an astroid (see accompanying figure). Find the area of the surface generated by revolving the curve about the x-axis.

Problem 24AAE Finding surface area Find the area of the surface generated by revolving the curve in Exercise 23 about the y-axis. Reference: Exercise 23 Length of a curve Find the length of the curve from y = 1n x from x = 1 to x = e.

Problem 26AAE Applications Length of a curve Find the length of the curve

Problem 25PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 26PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 27AAE Applications For what value or values of a does converge? Evaluate the corresponding integral(s).

Problem 27PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 28PE Partial Fractions Evaluate the integrals. It may be necessary to use a substitution first.

Problem 28AAE Applications For x > 0, each let Prove that xG(x) = 1 for each x > 0

Problem 29AAE Applications Infinite area and finite volume What values of p have the following property: The area of the region between the curve and the x-axis is infinite but the volume of the solid generated by revolving the region about the x-axis is finite.

Problem 29PE Trigonometric Substitutions Evaluate the integrals (a) without using a trigonometric substitution, (b) using a trigonometric substitution.

Problem 30AAE Applications Infinite area and finite volume What values of p have the following property: The area of the region in the first quadrant enclosed by the curve y = x-p ,the y-axis, the line x = 1, and the interval [0, 1] on the x-axis is infinite but the volume of the solid generated by revolving the region about one of the coordinate axes is finite.

Problem 30PE Trigonometric Substitutions Evaluate the integrals (a) without using a trigonometric substitution, (b) using a trigonometric substitution.

Problem 31AAE Integrating the square of the derivative If f is continuously differentiable on [0,1] and f(1) = f(0) = –1/6, prove that

Problem 32AAE (Continuation of exercise 31.) If f is continuously differentiable on [0,a] for a > 0, and f(a) = f(0) = b, prove that Reference: Integrating the square of the derivative If f is continuously differentiable on [0,1] and f(1) = f(0) = –1/6, prove that

Problem 31PE Trigonometric Substitutions Evaluate the integrals (a) without using a trigonometric substitution, (b) using a trigonometric substitution.

Problem 32PE Trigonometric Substitutions Evaluate the integrals (a) without using a trigonometric substitution, (b) using a trigonometric substitution.

Problem 33PE Trigonometric Substitutions Evaluate the integrals

Problem 33AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 34PE Trigonometric Substitutions Evaluate the integrals

Problem 34AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 35PE Trigonometric Substitutions Evaluate the integrals

Problem 35AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 36AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 36PE Trigonometric Substitutions Evaluate the integrals

Problem 37PE Trigonometric Integrals Evaluate the integrals

Problem 38AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 37AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 39PE Trigonometric Integrals Evaluate the integrals

Problem 39AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 38PE Trigonometric Integrals Evaluate the integrals

Problem 40AAE Tabular Integration The technique of tabular integration also applies to integrals of the form when neither function can be differentiated repeatedly to become zero. For example, to evaluate we begin as before with a table listing successive derivatives of e2x and integrals of cos x: We stop differentiating and integrating as soon as we reach a row that is the same as the first row except for multiplicative constants. We interpret the table as saying We take signed products from the diagonal arrows and a signed integral for the last horizontal arrow. Transposing the integral on the right-hand side over to the left-hand side now gives Or after dividing by 5 and adding the constant of integration. Use tabular integration to evaluate the integrals

Problem 41AAE The Substitution z = tan (x/2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 40PE Trigonometric Integrals Evaluate the integrals

Problem 41PE Trigonometric Integrals Evaluate the integrals

Problem 42PE Trigonometric Integrals Evaluate the integrals Step-by-step solution

Problem 43PE Trigonometric Integrals Evaluate the integrals

Problem 44AAE The Substitution z = tan (x>2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 44PE Trigonometric Integrals Evaluate the integrals

Problem 43AAE The Substitution z = tan (x>2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 45AAE The Substitution z = tan (x>2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 42AAE The Substitution z = tan (x>2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 45PE Numerical Integration According to the error-bound formula for Simpson’s Rule, how many subintervals should you use to be sure of estimating the value of by Simpson’s Rule with an error of no more than 10-4 in absolute value? (Remember that for Simpson’s Rule, the number of subintervals has to be even.)

Problem 46AAE The Substitution z = tan (x>2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 46PE Numerical Integration A brief calculation shows that if 0 ? x? 1, then the second derivative of lies between 0 and 8. Based on this, about how many subdivisions would you need to estimate the integral of ƒ from 0 to 1 with an error no greater than 10-3 in absolute value using the Trapezoidal Rule?

Problem 47AAE The Substitution z = tan (x>2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 48PE Numerical Integration You are planning to use Simpson’s Rule to estimate the value of the integral with an error magnitude less than 10–5.You have determined that throughout the interval of integration. How many subintervals should you use to assure the required accuracy? (Remember that for Simpson’s Rule the number has to be even.)

Problem 48AAE The Substitution z = tan (x>2) The substitution (7) reduces the problem of integrating a rational expression in sin x and cos x to a problem of integrating a rational function of z. This in turn can be integrated by partial fractions. From the accompanying figure we can read the relation Use the substitutions in Equations (7)–(10) to evaluate the integrals .Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

Problem 49AAE Use the substitution to evaluate the integrals

Problem 49PE Numerical Integration Mean temperature Use Simpson’s Rule to approximate the average value of the temperature function for a 365-day year. This is one way to estimate the annual mean air temperature in Fairbanks, Alaska. The National Weather Service’s official figure, a numerical average of the daily normal mean air temperatures for the year, is 25.7°F, which is slightly higher than the average value of ƒ(x).

Problem 50AAE Use the substitution to evaluate the integrals

Problem 47PE Numerical Integration A direct calculation shows that How close do you come to this value by using the Trapezoidal Rule with n = 6? Simpson’s Rule with n = 6? Try them and find out.

Problem 50PE Numerical Integration Heat capacity of a gas Heat capacity Cv is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal deg-mol (calories per degree gram molecular weight). The heat capacity of oxygen depends on its temperature T and satisfies the formula C v = 8.27 + 10–5 (26 T – 1.87T2) Use Simpson’s Rule to find the average value of and the temperature at which it is attained for 20 ? T ? 675?.

Problem 51PE Numerical Integration Fuel efficiency An automobile computer gives a digital readout of fuel consumption in gallons per hour. During a trip, a passenger recorded the fuel consumption every 5 min for a full hour of travel. a. Use the Trapezoidal Rule to approximate the total fuel consumption during the hour. b. If the automobile covered 60 mi in the hour, what was its fuel efficiency (in miles per gallon) for that portion of the trip?

Problem 51AAE The Gamma Function and Stirling’s Formula Euler’s gamma function (“gamma of x”; is a Greek capital g) uses an integral to extend the factorial function from the nonnegative integers to other real values. The formula is For each positive x, the number is the integral of t x-1e-t with respect to t from 0 to ?Figure 8.21 shows the graph of near the origin. You will see how to calculate if you do Additional Exercise 23 in Chapter 14. FIGURE 8.21 Euler’s gamma function is a continuous function of x whose value at each positive integer n + 1 is n!.The defining integral formula for is valid only for x > 0, but we can extend to negative noninteger values of x with the formula which is the subject of Exercise 31 If n is a nonnegative integer, a. Show that b. Then apply integration by parts to the integral for to show that This gives c. Use mathematical induction to verify Equation (1) for every nonnegative integer n.

Problem 52AAE The Gamma Function and Stirling’s Formula. Stirling’s formula Scottish mathematician James Stirling (1692–1770) showed that so, for large x, Dropping leads to the approximation a. Stirling’s approximation for n ! Use Equation (3) and the fact that to show that As you will see if you do Exercise 104 in Section 10.1, Equation (4) leads to the approximation(5) b. Compare your calculator’s value for n! with the value given by Stirling’s approximation for n = 10, 20, 30,…., as far as your calculator can go. c. A refinement of Equation (2) gives Compare the values given for 10! by your calculator, Stirling’s approximation, and Equation (6).

Problem 52PE Numerical Integration A new parking lot To meet the demand for parking, your town has allocated the area shown here. As the town engineer, you have been asked by the town council to find out if the lot can be built for $11,000. The cost to clear the land will be $0.10 a square foot, and the lot will cost $2.00 a square foot to pave. Use Simpson’s Rule to find out if the job can be done for $11,000.

Evaluate the improper integrals in Exercises 53–62. \(\int_{0}^{1} \ln x d x\) Equation Transcription: Text Transcription: \int_{0}^{1} \ln x d x

Problem 53PE Improper Integrals Evaluate the improper integrals

Problem 55PE Improper Integrals Evaluate the improper integrals

Problem 56PE Improper Integrals Evaluate the improper integrals

Problem 57PE Improper Integrals Evaluate the improper integrals

Problem 58PE Improper Integrals Evaluate the improper integrals

Problem 59PE Improper Integrals Evaluate the improper integrals

Problem 61PE Improper Integrals Evaluate the improper integrals

Problem 64PE Improper Integrals Which of the improper integrals converge and which diverge?

Problem 60PE Improper Integrals Evaluate the improper integrals

Problem 62PE Improper Integrals Evaluate the improper integrals

Problem 63PE Improper Integrals Which of the improper integrals converge and which diverge?

Problem 65PE Improper Integrals Which of the improper integrals converge and which diverge?

Problem 66PE Improper Integrals Which of the improper integrals converge and which diverge?

Problem 68PE Improper Integrals Which of the improper integrals converge and which diverge? d

Problem 70PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 67PE Improper Integrals Which of the improper integrals converge and which diverge?

Problem 69PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 71PE Evaluate the integrals in exercise. The integrals are listed in random order so you need to decide which integration technique to use.

Problem 72PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 73PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 74PE Evaluate the integrals in exercise. The integrals are listed in random order so you need to decide which integration technique to use.

Problem 75PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 76PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 78PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 79PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 77PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 81PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 80PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 84PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 82PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 83PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 86PE Evaluate the integrals in exercise. The integrals are listed in random order so you need to decide which integration technique to use.

Problem 87PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 88PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 91PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 89PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 90PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 85PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 93PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 92PE Evaluate the integrals in exercise. The integrals are listed in random order so you need to decide which integration technique to use.

Problem 94PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 95PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 96PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 97PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 98PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 99PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 100PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 101PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 102PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 104PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 103PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 105PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 106PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 107PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 108PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 109PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 111PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 112PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 110PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 113PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order a. Show that . b. Use part (a) to evaluate

Problem 114PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order

Problem 115PE Assorted Integrations Evaluate the integrals. The integrals are listed in random order