Chapter 5

Note: for this problem, because later answers depend on earlier ones, you must enter answers for all answer blanks for the problem to be correctly graded. If you would like to get feedback before you completed all computations, enter a "1" for each answer you did not yet compute and then submit the problem. (But note that this will, obviously, result in a problem submission.) (a) What is the exact value of \(\int_{0}^{3} e^{x} d x ?\) \(\int_{0}^{3} e^{x} d x=\) _______ Find LEFT(2), RIGHT(2), TRAP(2), MID(2), and SIMP(2); compute the error for each. $$ \begin{array}{|c|c|c|c|c|c|} \hline & \text { LEFT(2) } & \text { RIGHT(2) } & \text { TRAP(2) } & \text { MID(2) } & \text { SIMP(2) } \\ \hline \text { value } & & & & & \\ \hline \text { error } & & & & & \\ \hline \end{array} $$ (c) Repeat part (b) with \(n=4\) (instead of \(n=2\) ). $$ \begin{array}{|c|c|c|c|c|c|} \hline & \text { LEFT(4) } & \text { RIGHT(4) } & \text { TRAP(4) } & \text { MID(4) } & \text { SIMP(4) } \\ \hline \text { value } & & & & & \\ \hline \text { error } & & & & & \\ \hline \end{array} $$ \((d)\) For each rule in part \((\mathrm{b})\), as \(n\) goes from \(n=2\) to \(n=4\), does the error go down approximately as you would expect? Explain by calculating the ratios of the errors: Error LEFT(2)/Error LEFT(4) = Error RIGHT(2)/Error RIGHT(4)= Error TRAP(2)/Error TRAP(4) = Error \(\mathrm{MID}(2) /\) Error \(\mathrm{MID}(4)=\) Error \(\operatorname{SIMP}(2) /\) Error \(\operatorname{SIMP}(4)=\) (Be sure that you can explain in words why these do (or don't) make sense.)

Calculate the integral below by partial fractions and by using the indicated substitution. Be sure that you can show how the results you obtain are the same. $$ \int \frac{2 x}{x^{2}-25} d x $$ First, rewrite this with partial fractions: \(\int \frac{2 x}{x^{2}-25} d x=\int\) _____________ \(d x+\int\) _______________ \(d x=\) ______________ + ______________ \(+C\) (Note that you should not include the +C in your entered answer, as it has been provided at the end of the expression.) Next, use the substitution \(w=x^{2}-25\) to find the integral: \(\int \frac{2 x}{x^{2}-25} d x=\int\) ______________ \(d w=\) ________________ \(+C=\) _______________ \(+C\) (For the second answer blank, give your antiderivative in terms of the variable \(w .\) Again, note that you should not include the +C in your answer.)

For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integrals. 1\. \(\int x \sin x d x\) 2\. \(\int \frac{x^{2}}{1+x^{3}} d x\) 3\. \(\int x^{2} e^{x^{3}} d x\) 4\. \(\int x^{2} \cos \left(x^{3}\right) d x\) 5\. \(\int \frac{1}{\sqrt{3 x+1}} d x\) (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.)

Find the following integral. Note that you can check your answer by differentiation. \(\int t^{3}\left(t^{4}-3\right)^{3} d t=\) ______________________

Calculate the integral: \(\int \frac{1}{(x+6)(x+8)} d x=\) ___________________

Use integration by parts to evaluate the integral. \(\int 3 x \cos (2 x) d x=\) ___________

Find the the general antiderivative \(F(x)\) of the function \(f(x)\) given below. Note that you can check your answer by differentiation. \(f(x)=2 x^{3} \sin \left(x^{4}\right)\) antiderivative \(F(x)=\) ___________________________

Calculate the integral \(\int \frac{7 x+3}{x^{2}-3 x+2} d x=\) ________________

Find the integral \(\int(z+1) e^{4 z} d z=\) ___________

Find the following integral. Note that you can check your answer by differentiation. \(\int \frac{\ln ^{7}(z)}{z} d z=\) _______________________

Find a good numerical approximation to \(F(9)\) for the function with the properties that \(F^{\prime}(x)=e^{-x^{2} / 5}\) and \(F(0)=2\) \(F(9) \approx\) ___________.

Evaluate the definite integral. \(\int_{0}^{4} t e^{-t} d t=\) __________

The form of the partial fraction decomposition of a rational function is given below. \(\frac{25 x-10 x^{2}-45}{(x-5)\left(x^{2}+9\right)}=\frac{A}{x-5}+\frac{B x+C}{x^{2}+9}\) \(A=\) ______________ \(B=\) _____________ \(C=\) ______________ Now evaluate the indefinite integral. \(\int \frac{25 x-10 x^{2}-45}{(x-5)\left(x^{2}+9\right)} d x=\) ____________________

Let \(f(t)=t e^{-2 t}\) and \(F(x)=\int_{0}^{x} f(t) d t\) a. Determine \(F^{\prime}(x)\). b. Use the First FTC to find a formula for \(F\) that does not involve an integral. c. Is \(F\) an increasing or decreasing function for \(x>0\) ? Why?

The tide removes sand from the beach at a small ocean park at a rate modeled by the function $$ R(t)=2+5 \sin \left(\frac{4 \pi t}{25}\right) $$ A pumping station adds sand to the beach at rate modeled by the function $$ S(t)=\frac{15 t}{1+3 t} $$ Both \(R(t)\) and \(S(t)\) are measured in cubic yards of sand per hour, \(t\) is measured in hours, and the valid times are \(0 \leq t \leq 6\). At time \(t=0\), the beach holds 2500 cubic yards of sand. a. What definite integral measures how much sand the tide will remove during the time period \(0 \leq t \leq 6\) ? Why? b. Write an expression for \(Y(x)\), the total number of cubic yards of sand on the beach at time \(x\). Carefully explain your thinking and reasoning. c. At what instantaneous rate is the total number of cubic yards of sand on the beach at time \(t=4\) changing? d. Over the time interval \(0 \leq t \leq 6\), at what time \(t\) is the amount of sand on the beach least? What is this minimum value? Explain and justify your answers fully.

For each of the following integrals involving rational functions, (1) use a CAS to find the partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting function without the assistance of technology; (3) use a CAS to evaluate the original integral to test and compare your result in (2). a. \(\int \frac{x^{3}+x+1}{x^{4}-1} d x\) b. \(\int \frac{x^{5}+x^{2}+3}{x^{3}-6 x^{2}+11 x-6} d x\) c. \(\int \frac{x^{2}-x-1}{(x-3)^{3}} d x\)

Use the Fundamental Theorem of Calculus to find \(\int_{5 \pi / 2}^{3 \pi} e^{\sin (q)} \cdot \cos (q) d q=\) _____________________

When an aircraft attempts to climb as rapidly as possible, its climb rate (in feet per minute) decreases as altitude increases, because the air is less dense at higher altitudes. Given below is a table showing performance data for a certain single engine aircraft, giving its climb rate at various altitudes, where \(c(h)\) denotes the climb rate of the airplane at an altitude \(h\). $$ \begin{array}{lllllllllll} \hline h \text { (feet) } & 0 & 1000 & 2000 & 3000 & 4000 & 5000 & 6000 & 7000 & 8000 & 9000 & 10,000 \\ \hline c(\mathrm{ft} / \mathrm{min}) & 925 & 875 & 830 & 780 & 730 & 685 & 635 & 585 & 535 & 490 & 440 \\ \hline \end{array} $$ Let a new function \(m,\) that also depends on \(h,\) (say \(y=m(h))\) measure the number of minutes required for a plane at altitude \(h\) to climb the next foot of altitude. a. Determine a similar table of values for \(m(h)\) and explain how it is related to the table above. Be sure to discuss the units on \(m\). b. Give a careful interpretation of a function whose derivative is \(m(h) .\) Describe what the input is and what the output is. Also, explain in plain English what the function tells us. c. Determine a definite integral whose value tells us exactly the number of minutes required for the airplane to ascend to 10,000 feet of altitude. Clearly explain why the value of this integral has the required meaning. d. Determine a formula for a function \(M(h)\) whose value tells us the exact number of minutes required for the airplane to ascend to \(h\) feet of altitude. e. Estimate the values of \(M(6000)\) and \(M(10000)\) as accurately as you can. Include units on your results.

For an unknown function \(f(x)\), the following information is known. \- \(f\) is continuous on [3,6]\(;\) \- \(f\) is either always increasing or always decreasing on [3,6]\(;\) \- \(f\) has the same concavity throughout the interval [3,6]\(;\) \- As approximations to \(\int_{3}^{6} f(x) d x, L_{4}=7.23, R_{4}=6.75,\) and \(M_{4}=7.05 .\) a. Is \(f\) increasing or decreasing on [3,6]\(?\) What data tells you? b. Is \(f\) concave up or concave down on [3,6] ? Why? c. Determine the best possible estimate you can for \(\int_{3}^{6} f(x) d x,\) based on the given information

The rate at which water flows through Table Rock Dam on the White River in Branson, MO, is measured in thousands of cubic feet per second (TCFS). As engineers open the floodgates, flow rates are recorded according to the following chart. $$ \begin{array}{llllllll} \hline \text { seconds, } t & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline \text { flow in TCFS, } r(t) & 2000 & 2100 & 2400 & 3000 & 3900 & 5100 & 6500 \\ \hline \end{array} $$ a. What definite integral measures the total volume of water to flow through the dam in the 60 second time period provided by the table above? b. Use the given data to calculate \(M_{n}\) for the largest possible value of \(n\) to approximate the integral you stated in (a). Do you think \(M_{n}\) over- or under-estimates the exact value of the integral? Why? c. Approximate the integral stated in (a) by calculating \(S_{n}\) for the largest possible value of \(n,\) based on the given data. d. Compute \(\frac{1}{60} S_{n}\) and \(\frac{2000+2100+2400+3000+3900+5100+6500}{7} .\) What quantity do both of these values estimate? Which is a more accurate approximation?

For each of the following integrals involving radical functions, (1) use an appropriate \(u\) -substitution along with Appendix A to evaluate the integral without the assistance of technology, and (2) use a CAS to evaluate the original integral to test and compare your result in (1). a. \(\int \frac{1}{x \sqrt{9 x^{2}+25}} d x\) b. \(\int x \sqrt{1+x^{4}} d x\) c. \(\int e^{x} \sqrt{4+e^{2 x}} d x\) d. \(\int \frac{\tan (x)}{\sqrt{9-\cos ^{2}(x)}} d x\)

For each of the following indefinite integrals, determine whether you would use \(u\) -substitution, integration by parts, neither*, or both to evaluate the integral. In each case, write one sentence to explain your reasoning, and include a statement of any substitutions used. (That is, if you decide in a problem to let \(u=e^{3 x}\), you should state that, as well as that \(\left.d u=3 e^{3 x} d x .\right)\) Finally, use your chosen approach to evaluate each integral. (* one of the following problems does not have an elementary antiderivative and you are not expected to actually evaluate this integral; this will correspond with a choice of "neither" among those given.) a. \(\int x^{2} \cos \left(x^{3}\right) d x\) b. \(\int x^{5} \cos \left(x^{3}\right) d x\left(\right.\) Hint: \(\left.x^{5}=x^{2} \cdot x^{3}\right)\) c. \(\int x \ln \left(x^{2}\right) d x\) d. \(\int \sin \left(x^{4}\right) d x\) e. \(\int x^{3} \sin \left(x^{4}\right) d x\) f. \(\int x^{7} \sin \left(x^{4}\right) d x\)

This problem centers on finding antiderivatives for the basic trigonometric functions other than \(\sin (x)\) and \(\cos (x)\). a. Consider the indefinite integral \(\int \tan (x) d x\). By rewriting the integrand as \(\tan (x)=\) \(\frac{\sin (x)}{\cos (x)}\) and identifying an appropriate function-derivative pair, make a \(u\) -substitution and hence evaluate \(\int \tan (x) d x\). b. In a similar way, evaluate \(\int \cot (x) d x\). c. Consider the indefinite integral $$ \int \frac{\sec ^{2}(x)+\sec (x) \tan (x)}{\sec (x)+\tan (x)} d x $$ Evaluate this integral using the substitution \(u=\sec (x)+\tan (x)\). d. Simplify the integrand in (c) by factoring the numerator. What is a far simpler way to write the integrand? e. Combine your work in (c) and (d) to determine \(\int \sec (x) d x\). f. Using \((\mathrm{c})-(\mathrm{e})\) as a guide, evaluate \(\int \csc (x) d x\).

Consider the indefinite integral given by $$ \int \frac{\sqrt{x+\sqrt{1+x^{2}}}}{x} d x $$ a. Explain why \(u\) -substitution does not offer a way to simplify this integral by discussing at least two different options you might try for \(u\). b. Explain why integration by parts does not seem to be a reasonable way to proceed, either, by considering one option for \(u\) and \(d v\). c. Is there any line in the integral table in Appendix Athat is helpful for this integral? d. Evaluate the given integral using WolframAlpha. What do you observe?

Consider the indefinite integral \(\int x \sqrt{x-1} d x\). a. At first glance, this integrand may not seem suited to substitution due to the presence of \(x\) in separate locations in the integrand. Nonetheless, using the composite function \(\sqrt{x-1}\) as a guide, let \(u=x-1\). Determine expressions for both \(x\) and \(d x\) in terms of \(u .\) b. Convert the given integral in \(x\) to a new integral in \(u\). c. Evaluate the integral in (b) by noting that \(\sqrt{u}=u^{1 / 2}\) and observing that it is now possible to rewrite the integrand in \(u\) by expanding through multiplication. d. Evaluate each of the integrals \(\int x^{2} \sqrt{x-1} d x\) and \(\int x \sqrt{x^{2}-1} d x\). Write a paragraph to discuss the similarities among the three indefinite integrals in this problem and the role of substitution and algebraic rearrangement in each.

Consider the indefinite integral \(\int \sin ^{3}(x) d x\). a. Explain why the substitution \(u=\sin (x)\) will not work to help evaluate the given integral. b. Recall the Fundamental Trigonometric Identity, which states that \(\sin ^{2}(x)+\cos ^{2}(x)=1\). By observing that \(\sin ^{3}(x)=\sin (x) \cdot \sin ^{2}(x),\) use the Fundamental Trigonometric Identity to rewrite the integrand as the product of \(\sin (x)\) with another function. c. Explain why the substitution \(u=\cos (x)\) now provides a possible way to evaluate the integral in (b). d. Use your work in (a)-(c) to evaluate the indefinite integral \(\int \sin ^{3}(x) d x\). e. Use a similar approach to evaluate \(\int \cos ^{3}(x) d x\).

For the town of Mathland, MI, residential power consumption has shown certain trends over recent years. Based on data reflecting average usage, engineers at the power company have modeled the town's rate of energy consumption by the function $$ r(t)=4+\sin (0.263 t+4.7)+\cos (0.526 t+9.4) $$ Here, \(t\) measures time in hours after midnight on a typical weekday, and \(r\) is the rate of consumption in megawatts \(^{2}\) at time \(t\). Units are critical throughout this problem. a. Sketch a carefully labeled graph of \(r(t)\) on the interval [0,24] and explain its meaning. Why is this a reasonable model of power consumption? b. Without calculating its value, explain the meaning of \(\int_{0}^{24} r(t) d t .\) Include appropriate units on your answer. c. Determine the exact amount of power Mathland consumes in a typical day. d. What is Mathland's average rate of energy consumption in a given 24 -hour period? What are the units on this quantity?