Answer: In Exercises 41 46, solve the differential equation. Use a graphing utility to

In Exercises 1–26, find the indefinite integral. \(\int \frac{5}{x} d x\) Text Transcription: int 5/x dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{10}{x} d x\) Text Transcription: int 10/x dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{1}{x+1} d x\) Text Transcription: int 1/x+1 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{1}{x-5} d x\) Text Transcription: int 1/x-5 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{1}{2 x+5} d x\) Text Transcription: int 1/2x+5 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{1}{4-3 x} d x\) Text Transcription: int 1/4-3x dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x}{x^{2}-3} d x\) Text Transcription: int x/x^2-3 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{2}}{5-x^{3}} d x\) Text Transcription: int x^2/5-x^3 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{4 x^{3}+3}{x^{4}+3 x} d x\) Text Transcription: int 4x^3+3/^4+3x dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{2}-2 x}{x^{3}-3 x^{2}} d x\) Text Transcription: int x^2-2 x/x^3-3x^2 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{2}-4}{x} d x\) Text Transcription: int x^2-4/x dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x}{\sqrt{9-x^{2}}} d x\) Text Transcription: int x/sqrt 9-x^2 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{2}+2 x+3}{x^{3}+3 x^{2}+9 x} d x\) Text Transcription: int x^2+2x+3/x^3+3x^2+9x dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x(x+2)}{x^{3}+3 x^{2}-4} d x\) Text Transcription: int x(x+2)/x^3+3x^2-4 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{2}-3 x+2}{x+1} d x\) Text Transcription: int x^2-3x+2/x+1 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{2 x^{2}+7 x-3}{x-2} d x\) Text Transcription: int 2x^2+7x-/x-2 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{3}-3 x^{2}+5}{x-3} d x\) Text Transcription: int x^3-3x^2+5/x-3 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{3}-6 x-20}{x+5} d x\) Text Transcription: int x^3-6x-20/x+5 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{4}+x-4}{x^{2}+2} d x\) Text Transcription: int x^4+x-4/x^2+2 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x^{3}-3 x^{2}+4 x-9}{x^{2}+3} d x\) Text Transcription: int x^3-3x^2+4x-9/x^2+3 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{(\ln x)^{2}}{x} d x\) Text Transcription: int (ln x)^2/x dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{1}{x \ln x^{3}} d x\) Text Transcription: int 1/x ln x^3 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{1}{\sqrt{x+1}} d x\) Text Transcription: int 1/sqrt x+1 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{1}{x^{2 / 3}\left(1+x^{1 / 3}\right)} d x\) Text Transcription: int 1/x^2/3(1+x^1/3) dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{2 x}{(x-1)^{2}} d x\) Text Transcription: int 2 x/(x-1)^2 dx

In Exercises 1–26, find the indefinite integral. \(\int \frac{x(x-2)}{(x-1)^{3}} d x\) Text Transcription: int x(x-2)/(x-1)^3 dx

In Exercises 27-30, find the indefinite integral by u-substitution. (Hint: Let u be the denominator of the integrand.) \(\int \frac{1}{1+\sqrt{2 x}} d x\) Text Transcription: int 1/1+sqrt 2x dx

In Exercises 27-30, find the indefinite integral by u-substitution. (Hint: Let u be the denominator of the integrand.) \(\int \frac{1}{1+\sqrt{3 x}} d x\) Text Transcription: int 1/1+sqrt 3x dx

In Exercises 27-30, find the indefinite integral by u-substitution. (Hint: Let u be the denominator of the integrand.) \(\int \frac{\sqrt{x}}{\sqrt{x}-3} d x\) Text Transcription: int sqrt x/sqrt x-3 dx

In Exercises 27-30, find the indefinite integral by u-substitution. (Hint: Let u be the denominator of the integrand.) \(\int \frac{\sqrt[3]{x}}{\sqrt[3]{x}-1} d x\) Text Transcription: int sqrt3 x/sqrt3 x-1 dx

In Exercises 31-40, find the indefinite integral. \(\int \cot \frac{\theta}{3} d \theta\) Text Transcription: int cot theta/3 d theta

In Exercises 31-40, find the indefinite integral. \(\int \tan 5 \theta d \theta\) Text Transcription: int tan 5 theta d theta

In Exercises 31-40, find the indefinite integral. \(\int \csc 2 x d x\) Text Transcription: int csc 2x dx

In Exercises 31-40, find the indefinite integral. \(\int \sec \frac{x}{2} d x\) Text Transcription: int sec x/2 dx

In Exercises 31-40, find the indefinite integral. \(\int(\cos 3 \theta-1) d \theta\) Text Transcription: int (cos 3 theta-1) d theta

In Exercises 31-40, find the indefinite integral. \(\int\left(2-\tan \frac{\theta}{4}\right) d \theta\) Text Transcription: int (2-tan theta/4) d theta

In Exercises 31-40, find the indefinite integral. \(\int \frac{\cos t}{1+\sin t} d t\) Text Transcription: int cos t/1+sin t dt

In Exercises 31-40, find the indefinite integral. \(\int \frac{\csc ^{2} t}{\cot t} d t\) Text Transcription: int csc^2 t/cot t dt

In Exercises 31-40, find the indefinite integral. \(\int \frac{\sec x \tan x}{\sec x-1} d x\) Text Transcription: int sec x tan x/sec x-1 dx

In Exercises 31-40, find the indefinite integral. \(\int(\sec 2 x+\tan 2 x) d x\) Text Transcription: int(sec 2x+tan 2x) dx

In Exercises 41– 46, solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. \(\frac{d y}{d x}=\frac{4}{x}\), (1,2) Text Transcription: dy/d x=4/x

In Exercises 41– 46, solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. \(\frac{d y}{d x}=\frac{x-2}{x}\), (-1,0) Text Transcription: dy/dx=x-2/x

In Exercises 41– 46, solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. \(\frac{d y}{d x}=\frac{3}{2-x}\), (1,0) Text Transcription: dy/dx=3/2-x

In Exercises 41– 46, solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. \(\frac{d y}{d x}=\frac{2 x}{x^{2}-9}\), (0,4) Text Transcription: dy/dx=2x/x^2-9

In Exercises 41– 46, solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. \(\frac{d s}{d \theta}=\tan 2 \theta\), (0,2) Text Transcription: ds/d theta=tan 2 theta

In Exercises 41– 46, solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. \(\frac{d r}{d t}=\frac{\sec ^{2} t}{\tan t+1}\), \((\pi, 4)\) Text Transcription: dr/dt=sec^2 t/tan t+1 (pi, 4)

Determine the function f if \(f^{\prime \prime}(x)=\frac{2}{x^{2}}\), f(1)=1, \(f^{\prime}(1)=1\), x>0 . Text Transcription: f^prime prime(x)=2/x^2 f^prime(1)=1

Determine the function f if \(f^{\prime \prime}(x)=-\frac{4}{(x-1)^{2}}-2\), f(2)=3, \(f^{\prime}(2)=0\), x>1 Text Transcription: f^prime prime(x)=-4/(x-1)^2-2 f^prime(2)=0

Slope Fields In Exercises 49-52, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(\frac{d y}{d x}=\frac{1}{x+2}\), (0,1) Text Transcription: dy/dx=1/x+2

Slope Fields In Exercises 49-52, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(\frac{d y}{d x}=\frac{\ln x}{x}\), (1,-2) Text Transcription: dy/d x=ln x/x

Slope Fields In Exercises 49-52, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(\frac{d y}{d x}=1+\frac{1}{x}\), (1,4) Text Transcription: dy/dx=1+1/x

Slope Fields In Exercises 49-52, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(\frac{d y}{d x}=\sec x\), (0,1) Text Transcription: dy/dx=sec x

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{4} \frac{5}{3 x+1} d x\) Text Transcription: int_0^4 5/3x+1 dx

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{-1}^{1} \frac{1}{2 x+3} d x\) Text Transcription: int_-1^1 1/2x+3 dx

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{1}^{e} \frac{(1+\ln x)^{2}}{x} d x\) Text Transcription: int_1^e (1+ln x)^2/x dx

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{e}^{e^{2}} \frac{1}{x \ln x} d x\) Text Transcription: int_e^e^2 1/x ln x dx

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{2} \frac{x^{2}-2}{x+1} d x\) Text Transcription: int_0^2 x^2-2/x+1 dx

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{1} \frac{x-1}{x+1} d x\) Text Transcription: int_0^1 x-1/x+1 dx

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{1}^{2} \frac{1-\cos \theta}{\theta-\sin \theta} d \theta\) Text Transcription: int_1^2 1-cos theta/theta-sin theta d theta

In Exercises 53– 60, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0.1}^{0.2}(\csc 2 \theta-\cot 2 \theta)^{2} d \theta\) Text Transcription: int_0.1^0.2(csc 2 theta-cot 2 theta)^2 d theta

In Exercises 61-66, use a computer algebra system to find or evaluate the integral. \(\int \frac{1}{1+\sqrt{x}} d x\) Text Transcription: int 1/1+sqrt x dx

In Exercises 61-66, use a computer algebra system to find or evaluate the integral. \(\int \frac{1-\sqrt{x}}{1+\sqrt{x}} d x\) Text Transcription: int 1-sqrt x/1+sqrt x dx

In Exercises 61-66, use a computer algebra system to find or evaluate the integral. \(\int \frac{\sqrt{x}}{x-1} d x\) Text Transcription: int sqrt x/x-1 dx

In Exercises 61-66, use a computer algebra system to find or evaluate the integral. \(\int \frac{x^{2}}{x-1} d x\) Text Transcription: int x^2/x-1 dx

In Exercises 61-66, use a computer algebra system to find or evaluate the integral. \(\int_{\pi / 4}^{\pi / 2}(\csc x-\sin x) d x\) Text Transcription: int_pi/4^pi/2(csc x-sin x) dx

In Exercises 61-66, use a computer algebra system to find or evaluate the integral. \(\int_{-\pi / 4}^{\pi / 4} \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x\) Text Transcription: int_-pi/4^pi/4 sin^2 x-cos^2 x/cos x dx

In Exercises 67-70, find \(F^{\prime}(x)\). \(F(x)=\int_{1}^{x} \frac{1}{t} d t\) Text Transcription: F^prime(x) F(x)=int_1^x 1/t dt

In Exercises 67-70, find \(F^{\prime}(x)\). \(F(x)=\int_{0}^{x} \tan t d t\) Text Transcription: F^prime(x) F(x)=int_0^x tan t dt

In Exercises 67-70, find \(F^{\prime}(x)\). \(F(x)=\int_{1}^{3 x} \frac{1}{t} d t\) Text Transcription: F^prime(x) F(x)=int_1^3x 1/t dt

In Exercises 67-70, find \(F^{\prime}(x)\). \(F(x)=\int_{1}^{x^{2}} \frac{1}{t} d t\) Text Transcription: F^prime(x) F(x)=int_1^x^2 1/t dt

Approximation In Exercises 71 and 72, determine which value best approximates the area of the region between the -axis and the graph of the function over the given interval. (Make your selection on the basis of a sketch of the region and not by performing any calculations.) f(x)=sec x, [0,1] (a) 6 (b) -6 (c) \(\frac{1}{2}\) (d) 1.25 (e) 3 Text Transcription: 1/2

Approximation In Exercises 71 and 72, determine which value best approximates the area of the region between the -axis and the graph of the function over the given interval. (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=\frac{2 x}{x^{2}+1}\), [0,4] (a) 3 (b) 7 (c) -2 (d) 5 (e) 1 Text Transcription: f(x)=2x/x^2+1

Area In Exercises 73–76, find the area of the given region. Use a graphing utility to verify your result. \(y=\frac{6}{x}\) Text Transcription: y=6/x

Area In Exercises 73–76, find the area of the given region. Use a graphing utility to verify your result. \(y=\frac{2}{x \ln x}\) Text Transcription: y=2/x ln x

Area In Exercises 73–76, find the area of the given region. Use a graphing utility to verify your result. y=tan x

Area In Exercises 73–76, find the area of the given region. Use a graphing utility to verify your result. \(y=\frac{\sin x}{1+\cos x}\) Text Transcription: y=sin x/1+cos x

Area In Exercises 77-80, find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. \(y=\frac{x^{2}+4}{x}\), x=1, x=4, y=0 Text Transcription: y=x^2+4/x

Area In Exercises 77-80, find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. \(y=\frac{x+6}{x}\), x=1, x=5, y=0 Text Transcription: y=x+6/x

Area In Exercises 77-80, find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. \(y=2 \sec \frac{\pi x}{6}\), x=0, x=2, y=0 Text Transcription: y=2 sec pi x/6

Area In Exercises 77-80, find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. y=2x-tan 0.3x, x=1, x=4, y=0

Numerical Integration In Exercises 81-84, use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let n=4 and round your answer to four decimal places. Use a graphing utility to verify your result. \(\int_{1}^{5} \frac{12}{x} d x\) Text Transcription: int_1^5 12/x dx

Numerical Integration In Exercises 81-84, use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let n=4 and round your answer to four decimal places. Use a graphing utility to verify your result. \(\int_{0}^{4} \frac{8 x}{x^{2}+4} d x\) Text Transcription: int_0^4 8 x/x^2+4 dx

Numerical Integration In Exercises 81-84, use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let n=4 and round your answer to four decimal places. Use a graphing utility to verify your result. \(\int_{2}^{6} \ln x d x\) Text Transcription: int_2^6 ln x dx

Numerical Integration In Exercises 81-84, use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let n=4 and round your answer to four decimal places. Use a graphing utility to verify your result. \(\int_{-\pi / 3}^{\pi / 3} \sec x d x\) Text Transcription: int_-pi/3^pi/3 sec x dx

WRITING ABOUT CONCEPTS In Exercises 85-88, state the integration formula you would use to perform the integration. Do not integrate. \(\int \sqrt[3]{x} d x\) Text Transcription: int sqrt3 x dx

WRITING ABOUT CONCEPTS In Exercises 85-88, state the integration formula you would use to perform the integration. Do not integrate. \(\int \frac{x}{\left(x^{2}+4\right)^{3}} d x\) Text Transcription: int x/(x^2+4)^3 dx

WRITING ABOUT CONCEPTS In Exercises 85-88, state the integration formula you would use to perform the integration. Do not integrate. \(\int \frac{x}{x^{2}+4} d x\) Text Transcription: int x/x^2+4 dx

WRITING ABOUT CONCEPTS In Exercises 85-88, state the integration formula you would use to perform the integration. Do not integrate. \(\int \frac{\sec ^{2} x}{\tan x} d x\) Text Transcription: int sec^2 x/tan x d x

Find a value of x such that \(\int_{1}^{x} \frac{3}{t} d t=\int_{1 / 4}^{x} \frac{1}{t} d t\). Text Transcription: int_1^x 3/t d t=int_1/4^x 1/t dt

CAPSTONE Find a value of x such that \(\int_{1}^{x} \frac{1}{t} d t\) is equal to (a) ln 5 and (b) 1 . Text Transcription: int_1^x 1/t dt

Show that \(\int \cot u d u=\ln |\sin u|+C\). Text Transcription: int cot u du=ln |sin u|+C

Show that \(\int \csc u d u=-\ln |\csc u+\cot u|+C\). Text Transcription: int csc u du=-ln |csc u+cot u|+C

In Exercises 93-96, show that the two formulas are equivalent. \(\int \tan x d x=-\ln |\cos x|+C\) \(\int \tan x d x=\ln |\sec x|+C\) Text Transcription: int tan x dx=-ln |cos x|+C int tan x dx=ln |sec x|+C

In Exercises 93-96, show that the two formulas are equivalent. \(\int \cot x d x=\ln |\sin x|+C\) \(\int \cot x d x=-\ln |\csc x|+C\) Text Transcription: int cot x dx=ln |sin x|+C int cot x dx=-ln |csc x|+C

In Exercises 93-96, show that the two formulas are equivalent. \(\int \sec x d x=\ln |\sec x+\tan x|+C\) \(\int \sec x d x=-\ln |\sec x-\tan x|+C\) Text Transcription: int sec x dx=ln |sec x+tan x|+C int sec x dx=-ln |sec x-tan x|+C

In Exercises 93-96, show that the two formulas are equivalent. \(\int \csc x d x=-\ln |\csc x+\cot x|+C\) \(\int \csc x d x=\ln |\csc x-\cot x|+C\) Text Transcription: int csc x dx=-ln |csc x+cot x|+C int csc x dx=ln |csc x-cot x|+C

In Exercises 97–100, find the average value of the function over the given interval. \(f(x)=\frac{8}{x^{2}}\), [2,4] Text Transcription: f(x)=8/x^2

In Exercises 97–100, find the average value of the function over the given interval. \(f(x)=\frac{4(x+1)}{x^{2}}\), [2,4] Text Transcription: f(x)=4(x+1)/x^2

In Exercises 97–100, find the average value of the function over the given interval. \(f(x)=\frac{2 \ln x}{x}\), [1, e] Text Transcription: f(x)=2 ln x/x

In Exercises 97–100, find the average value of the function over the given interval. \(f(x)=\sec \frac{\pi x}{6}\), [0,2] Text Transcription: f(x)=sec pi x/6

Population Growth A population of bacteria is changing at a rate of \(\frac{d P}{d t}=\frac{3000}{1+0.25 t}\) where t is the time in days. The initial population (when t=0 ) is 1000 . Write an equation that gives the population at any time t, and find the population when t=3 days. Text Transcription: dP/dt=3000/1+0.25 t

Heat Transfer Find the time required for an object to cool from \(300^{\circ} \mathrm{F}\) to \(250^{\circ} \mathrm{F}\) by evaluating \(t=\frac{10}{\ln 2} \int_{250}^{300} \frac{1}{T-100} d T\) where t is time in minutes. Text Transcription: 300^circ F 250^circ F t=10/ln 2 int_250^300 1/T-100 dT

Average Price The demand equation for a product is \(p=\frac{90,000}{400+3 x}\) . Find the average price p on the interval \(40 \leq x \leq 50\). Text Transcription: p=90,000/400+3x 40 leq x leq 50

Sales The rate of change in sales S is inversely proportional to time t(t>1) measured in weeks. Find S as a function of t if sales after 2 and 4 weeks are 200 units and 300 units, respectively.

True or False? In Exercises 105-108, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \((\ln x)^{1 / 2}=\frac{1}{2}(\ln x)\) Text Transcription: (ln x)^1/2=1/2(ln x)

True or False? In Exercises 105-108, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\int \ln x d x=(1 / x)+C\) Text Transcription: int ln x dx=(1/x)+C

True or False? In Exercises 105-108, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\int \frac{1}{x} d x=\ln |c x|\), \(c \neq 0\) Text Transcription: int 1/x dx=ln |c x| c neq 0

True or False? In Exercises 105-108, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\int_{-1}^{2} \frac{1}{x} d x=[\ln |x|]_{-1}^{2}=\ln 2-\ln 1=\ln 2\) Text Transcription: int_-1^2 1/x dx=[ln |x|]_-1^2=ln 2-ln 1=ln 2

Orthogonal Trajectory (a) Use a graphing utility to graph the equation \(2 x^{2}-y^{2}=8\). (b) Evaluate the integral to find \(y^{2}\) in terms of x. \(y^{2}=e^{-\int(1 / x) d x}\) For a particular value of the constant of integration, graph the result in the same viewing window used in part (a). (c) Verify that the tangents to the graphs in parts (a) and (b) are perpendicular at the points of intersection. Text Transcription: 2 x^2-y^2=8 y^2 y^2=e^-int(1/x) dx

Graph the function \(f(x)=\frac{x}{1+x^{2}}\) on the interval \([0, \infty)\). (a) Find the area bounded by the graph of f and the line \(y=\frac{1}{2} x\). (b) Determine the values of the slope m such that the line y=mx and the graph of f enclose a finite region. (c) Calculate the area of this region as a function of m. Text Transcription: f(x)=x/1+x^2 [0, infty) y=1/2 x

Napier's Inequality For 0<x<y, show that \(\frac{1}{y}<\frac{\ln y-\ln x}{y-x}<\frac{1}{x}\) Text Transcription: 1/y<ln y-ln x/y-x<1/x

Prove that the function \(F(x)=\int_{x}^{2 x} \frac{1}{t} d t\) is constant on the interval \((0, \infty)\). Text Transcription: F(x)=int_x^2x 1/t dt (0, infty)