University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 7

Integrals and Transcendental Functions

Educators


Problem 1

Evaluate the integrals.
$$\int_{-3}^{-2} \frac{d x}{x}$$

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Problem 2

Evaluate the integrals.
$$\int_{-1}^{0} \frac{3 d x}{3 x-2}$$

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Problem 3

Evaluate the integrals.
$$\int \frac{2 y d y}{y^{2}-25}$$

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Problem 4

Evaluate the integrals.
$$\int \frac{8 r d r}{4 r^{2}-5}$$

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Problem 5

Evaluate the integrals.
$$\int \frac{3 \sec ^{2} t}{6+3 \tan t} d t$$

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Problem 6

Evaluate the integrals.
$$\int \frac{\sec y \tan y}{2+\sec y} d y$$

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Problem 7

Evaluate the integrals.
$$\int \frac{d x}{2 \sqrt{x}+2 x}$$

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Problem 8

Evaluate the integrals.
$$\int \frac{\sec x d x}{\sqrt{\ln (\sec x+\tan x)}}$$

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Problem 9

Evaluate the integrals.
$$\int_{\ln 2}^{\ln 3} e^{x} d x$$

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Problem 10

Evaluate the integrals.
$$\int 8 e^{(x+1)} d x$$

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Problem 11

Evaluate the integrals.
$$\int_{1}^{4} \frac{(\ln x)^{3}}{2 x} d x$$

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Problem 12

Evaluate the integrals.
$$\int \frac{\ln (\ln x)}{x \ln x} d x$$

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Problem 13

Evaluate the integrals.
$$\int_{\ln 4}^{\ln 9} e^{x / 2} d x$$

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Problem 14

Evaluate the integrals.
$$\int \tan x \ln (\cos x) d x$$

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Problem 15

Evaluate the integrals.
$$\int \frac{e^{\sqrt{r}}}{\sqrt{r}} d r$$

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Problem 16

Evaluate the integrals.
$$\int \frac{e^{-\sqrt{r}}}{\sqrt{r}} d r$$

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Problem 17

Evaluate the integrals.
$$2 t e^{-t^{2}} d t$$

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Problem 18

Evaluate the integrals.
$$\int \frac{\ln x d x}{x \sqrt{\ln ^{2} x+1}}$$

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Problem 19

Evaluate the integrals.
$$\int \frac{e^{1 / x}}{x^{2}} d x$$

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Problem 20

Evaluate the integrals.
$$\int \frac{e^{-1 / x^{2}}}{x^{3}} d x$$

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Problem 21

Evaluate the integrals.
$$\int e^{\sec \pi t} \sec \pi t \tan \pi t d t$$

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Problem 22

Evaluate the integrals.
$$\int e^{\csc (\pi+t)} \csc (\pi+t) \cot (\pi+t) d t$$

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Problem 23

Evaluate the integrals.
$$\int_{\ln (\pi / 6)}^{\ln (\pi / 2)} 2 e^{v} \cos e^{v} d v$$

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Problem 24

Evaluate the integrals.
$$\int_{0}^{\sqrt{\ln \pi}} 2 x e^{x^{2}} \cos \left(e^{x^{2}}\right) d x$$

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Problem 25

Evaluate the integrals.
$$\int \frac{e^{r}}{1+e^{r}} d r$$

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Problem 26

Evaluate the integrals.
$$\int \frac{d x}{1+e^{x}}$$

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Problem 27

Evaluate the integrals.
$$\int_{0}^{1} 2^{-\theta} d \theta$$

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Problem 28

Evaluate the integrals.
$$\int_{-2}^{0} 5^{-\theta} d \theta$$

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Problem 29

Evaluate the integrals.
$$\int_{1}^{\sqrt{2}} x 2^{\left(x^{2}\right)} d x$$

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Problem 30

Evaluate the integrals.
$$\int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} d x$$

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Problem 31

Evaluate the integrals.
$$\int_{0}^{\pi / 2} 7^{\cos t} \sin t d t$$

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Problem 32

Evaluate the integrals.
$$\int_{0}^{\pi / 4}\left(\frac{1}{3}\right)^{\tan t} \sec ^{2} t d t$$

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Problem 33

Evaluate the integrals.
$$\int_{2}^{4} x^{2 x}(1+\ln x) d x$$

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Problem 34

Evaluate the integrals.
$$\int_{1}^{2} \frac{2^{\ln x}}{x} d x$$

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Problem 35

Evaluate the integrals.
$$\int_{0}^{3}(\sqrt{2}+1) x^{\sqrt{2}} d x$$

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Problem 36

Evaluate the integrals.
$$\int_{1}^{e} x^{(\ln 2)-1} d x$$

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Problem 37

Evaluate the integrals.
$$\int \frac{\log _{10} x}{x} d x$$

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Problem 38

Evaluate the integrals.
$$\int_{1}^{4} \frac{\log _{2} x}{x} d x$$

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Problem 39

Evaluate the integrals.
$$\int_{1}^{4} \frac{\ln 2 \log _{2} x}{x} d x$$

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Problem 40

Evaluate the integrals.
$$\int_{1}^{e} \frac{2 \ln 10 \log _{10} x}{x} d x$$

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Problem 41

Evaluate the integrals.
$$\int_{0}^{2} \frac{\log _{2}(x+2)}{x+2} d x$$

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Problem 42

Evaluate the integrals.
$$\int_{1 / 10}^{10} \frac{\log _{10}(10 x)}{x} d x$$

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Problem 43

Evaluate the integrals.
$$\int_{0}^{9} \frac{2 \log _{10}(x+1)}{x+1} d x$$

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Problem 44

Evaluate the integrals.
$$\int_{2}^{3} \frac{2 \log _{2}(x-1)}{x-1} d x$$

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Problem 45

Evaluate the integrals.
$$\int \frac{d x}{x \log _{10} x}$$

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Problem 46

Evaluate the integrals.
$$\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$$

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Problem 47

Solve the initial value.
$$\frac{d y}{d t}=e^{t} \sin \left(e^{t}-2\right), \quad y(\ln 2)=0$$

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Problem 48

Solve the initial value.
$$\frac{d y}{d t}=e^{-t} \sec ^{2}\left(\pi e^{-t}\right), \quad y(\ln 4)=2 / \pi$$

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Problem 49

Solve the initial value.
$$\frac{d^{2} y}{d x^{2}}=2 e^{-x}, \quad y(0)=1 \quad \text { and } \quad y^{\prime}(0)=0$$

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Problem 50

Solve the initial value.
$$\frac{d^{2} y}{d t^{2}}=1-e^{2 t}, \quad y(1)=-1 \quad \text { and } \quad y^{\prime}(1)=0$$

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Problem 51

Solve the initial value.
$$\frac{d y}{d x}=1+\frac{1}{x}, \quad y(1)=3$$

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Problem 52

Solve the initial value.
$$\frac{d^{2} y}{d x^{2}}=\sec ^{2} x, \quad y(0)=0 \quad \text { and } \quad y^{\prime}(0)=1$$

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Problem 53

The region between the curve $y=1 / x^{2}$ and the $x$ -axis from $x=1 / 2$ to $x=2$ is revolved about the $y$ -axis to generate a solid. Find the volume of the solid.

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Problem 54

In Section $6.2,$ Exercise $6,$ we revolved about the $y$ -axis the region between the curve $y=9 x / \sqrt{x^{3}+9}$ and the $x$ -axis from $x=0$
to $x=3$ to generate a solid of volume $36 \pi .$ What volume do you get if you revolve the region about the $x$ -axis instead?

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Problem 55

Find the lengths of the curves.
$$y=\left(x^{2} / 8\right)-\ln x, \quad 4 \leq x \leq 8$$

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Problem 56

Find the lengths of the curves.
$$x=(y / 4)^{2}-2 \ln (y / 4), \quad 4 \leq y \leq 12$$

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Problem 57

The linearization of $\ln (1+x)$ at $x=0$ Instead of approximating $\ln x$ near $x=1,$ we approximate $\ln (1+x)$ near $x=0$ We get a simpler formula this way.
a. Derive the linearization $\ln (1+x) \approx x$ at $x=0$
b. Estimate to five decimal places the error involved in replacing $\ln (1+x)$ by $x$ on the interval [0,0.1]
c. Graph $\ln (1+x)$ and $x$ together for $0 \leq x \leq 0.5 .$ Use different colors, if available. At what points does the approximation of $\ln (1+x)$ seem best? Least good? By reading coordinates from the graphs, find as good an upper bound for the error as your grapher will allow.

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Problem 58

The linearization of $e^{x}$ at $x=0$
a. Derive the linear approximation $e^{x} \approx 1+x$ at $x=0$
b. Estimate to five decimal places the magnitude of the error involved in replacing $e^{x}$ by $1+x$ on the interval [0,0.2]
c. Graph $e^{x}$ and $1+x$ together for $-2 \leq x \leq 2 .$ Use different colors, if available. On what intervals does the approximation appear to overestimate $e^{x}$ ? Underestimate $e^{x} ?$

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Problem 59

The geometric, logarithmic, and arithmetic mean inequality. (FIGURE CAN'T COPY)
a. Show that the graph of $e^{x}$ is concave up over every interval of $x$ -values.
b. Show, by reference to the accompanying figure, that if $0<a<b,$ then $e^{(\ln a+\ln b) / 2} \cdot(\ln b-\ln a)<\int_{\ln a}^{\ln b} e^{x} d x<\frac{e^{\ln a}+e^{\ln b}}{2} \cdot(\ln b-\ln a)$
c. Use the inequality in part (b) to conclude that $$\sqrt{a b}<\frac{b-a}{\ln b-\ln a}<\frac{a+b}{2}$$. This inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in turn is less than their arithmetic mean.

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Problem 60

Use Figure 7.1 and appropriate areas to show that $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}<\ln n<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n-1}$

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Problem 61

Partition the interval [1,2] into $n$ equal parts. Then use Figure 7.1 and appropriate partition points and areas to show that $$\begin{aligned}\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{2 n}<\ln 2<\frac{1}{n}+\frac{1}{n+1} \\+& \frac{1}{n+2}+\cdots+\frac{1}{2 n-1}\end{aligned}$$

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Problem 62

Could $x^{\ln 2}$ possibly be the same as $2^{\ln x}$ for some $x>0 ?$ Graph the two functions and explain what you see.

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Problem 63

Which is bigger, $\pi^{e}$ or $e^{\pi} ?$ Calculators have taken some of the mystery out of this once-challenging question. (Go ahead and check; you will see that it is a surprisingly close call.) You can answer the question without a calculator, though. (FIGURE CAN'T COPY)
a. Find an equation for the line through the origin tangent to the graph of $y=\ln x$
b. Give an argument based on the graphs of $y=\ln x$ and the tangent line to explain why $\ln x<x / e$ for all positive $x \neq e$
c. Show that $\ln \left(x^{e}\right)<x$ for all positive $x \neq e$
d. Conclude that $x^{e}<e^{x}$ for all positive $x \neq e$
e. So which is bigger, $\pi^{e}$ or $e^{\pi} ?$

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Problem 64

Find $e$ to as many decimal places as you can by solving the equation $\ln x=1$ using Newton's method in Section 4.7.

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Problem 65

Most scientific calculators have keys for $\log _{10} x$ and $\ln x .$ To find logarithms to other bases, we use the equation $\log _{a} x=$ $(\ln x) /(\ln a)$
Find the following logarithms to five decimal places.
a. $\log _{3} 8$
b. $\log _{7} 0.5$
c. $\log _{20} 17$
d. $\log _{0.5} 7$
e. $\ln x,$ given that $\log _{10} x=2.3$
f. $\ln x,$ given that $\log _{2} x=1.4$
g. $\ln x,$ given that $\log _{2} x=-1.5$
h. $\ln x,$ given that $\log _{10} x=-0.7$

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Problem 66

Conversion factors
a. Show that the equation for converting base 10 logarithms to base 2 logarithms is $$\log _{2} x=\frac{\ln 10}{\ln 2} \log _{10} x$$
b. Show that the equation for converting base $a$ logarithms to base $b$ logarithms is $$\log _{b} x=\frac{\ln a}{\ln b} \log _{a} x$$

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Problem 67

Alternative proof that $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$
a. Let $x>0$ be given, and use Figure 7.1 to show that $$\frac{1}{x+1}<\int_{x}^{x+1} \frac{1}{t} d t<\frac{1}{x}$$
b. Conclude from part (a) that $$\frac{1}{x+1}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x}$$
c. Conclude from part (b) that $$e^{\frac{x}{x+1}}<\left(1+\frac{1}{x}\right)^{x}<e$$
d. Conclude from part (c) that $$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$$

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University of North Texas