Question 1.

\(\dfrac{d}{dx}\left[\int f(x)\, dx\right] =\) _________

Question 2.

True or False  If \(k\) is a constant, then \[ \int kf(x)\, dx = \left[ \int kdx \right] \left[ \int f(x)\, dx\right] \]

385

Question 3.

If \(a\) is a number, then \(\int x^{a}\, dx=\) _________, provided \(a \neq -1\).

Question 4.

True or False  When integrating a function \(f\), a constant of integration \(C\) is added to the result because \(\int f(x)\, dx\) denotes all the antiderivatives of \(f\).

Question 5.

\(\int x^{2/3}{\,dx} \)

Question 6.

\(\int {t^{-4}\,dt} \)

Question 7.

\(\int \dfrac{1}{\sqrt{1-x^{2}}}\,dx \)

Question 8.

\(\int \dfrac{1}{1+x^{2}}\,dx\)

Question 9.

\(\int \dfrac{5x^{2}+2x-1}{x} \,dx\)

Question 10.

\(\int \dfrac{x+1}{x} \,dx\)

Question 11.

\(\int \dfrac{4}{3t}\,dt \)

Question 12.

\(\int 2{e}^{u}{du} \)

Question 13.

\(\int {(4x^{3}-3x^{2}+5x-2)\,dx}\)

Question 14.

\(\int {(3x^{5}-2x^{4}-x^{2}-1)\,dx}\)

Question 15.

\(\int \left( \dfrac{1}{x^{3}}+1\right) dx \)

Question 16.

\(\int \left( {x-{\dfrac{1}{x^{2}}}} \right)\, dx \)

Question 17.

\(\int {(3\sqrt{z}+z)\,dz} \)

Question 18.

\(\int {(4\sqrt{x}+1)\,dx}\)

Question 19.

\(\int {(4t^{3/2}+t^{1/2})\,dt}\)

Question 20.

\(\int \left({3x^{2/3}-\dfrac{1}{\sqrt{x}}} \right)\, dx\)

Question 21.

\(\int {u(u-1)\,du} \)

Question 22.

\(\int {t^{2}(t+1)\,dt}\)

Question 23.

\(\int {{\dfrac{{3x^{5}+1}}{{x^{2}}}}\,dx} \)

Question 24.

\(\int \dfrac{x^{2}+2x+1}{x^{4}}\,dx \)

Question 25.

\(\int \dfrac{t^{2}-4}{t-2}\,dt \)

Question 26.

\(\int {{\dfrac{{z^{2}-16}}{{z+4}}}dz}\)

Question 27.

\(\int {(2x+1)^{2}\,dx} \)

Question 28.

\(\int 3{(x^{2}\,{+}\,1)^{2}\,dx}\)

Question 29.

\(\int (x+e^{x})\, dx \)

Question 30.

\(\int (2e^{x}-x^{3})\, dx\)

Question 31.

\(\int 8(1+x^{2}) ^{-1}dx\)

Question 32.

\(\int \dfrac{-7}{1+x^{2}}\,dx\)

Question 33.

\(\int \dfrac{x^{2}-1}{2x^{3}}\,dx\)

Question 34.

\(\int \dfrac{x^{2}+4x-1}{x^{2}}\,dx\)

Question 35.

\(\int \dfrac{\tan x}{\cos x}\,dx\)

Question 36.

\(\int \dfrac{1}{\sin ^{2}x}\,dx\)

Question 37.

\(\int \dfrac{2}{5\sqrt{1-x^{2}}}\,dx \)

Question 38.

\(\int -\dfrac{4}{x\sqrt{x^{2}-1}}\,dx\)

Question 39.

\(\dfrac{dy}{dx}=e^{x}\), \(y=4\) when \(x=0\)

Question 40.

\(\dfrac{dy}{dx}=\dfrac{1}{x}\), \(y=0\) when \(x=1\)

Question 41.

\(\dfrac{dy}{dx}=\dfrac{x^{2}+x+1}{x}\), \(y=0\) when \(x=1\)

Question 42.

\(\dfrac{dy}{dx}=x+e^{x}\), \(y=4\) when \(x=0\)

Question 43.

\(\dfrac{dy}{dx}=xy^{1/2}\), \(y=1\) when \(x=2\)

Question 44.

\(\dfrac{dy}{dx}=x^{1/2}y\), \(y=1\) when \(x=0\)

Question 45.

\(\dfrac{dy}{dx}=\dfrac{y-1}{x-1} \), \(y=2\) when \(x=2 \)

Question 46.

\(\dfrac{dy}{dx}=\dfrac{y}{x}\), \(y=2\) when \(x=1\)

Question 47.

\(\dfrac{dy}{dx}=\dfrac{x}{\cos y},\) \(y=\pi \) when \(x=2\)

Question 48.

\(\dfrac{dy}{dx}=y\sin x,\) \(y=e\) when \(x=0\)

Question 49.

\(\dfrac{dy}{dx}=\dfrac{4e^{x}}{y},\) \(y=2\) when \(x=0\)

Question 50.

\(\dfrac{dy}{dx}=5ye^{x},\) \(y=1\) when \(x=0\)

Question 51.

Uninhibited Growth The population of a colony of mosquitoes obeys the uninhibited growth equation \(\dfrac{dN}{dt}=kN\). If there are \(1500\) mosquitoes initially, and there are \(2500\) mosquitoes after 24 h, what is the mosquito population after 3 days?

Question 52.

Radioactive Decay A radioactive substance follows the decay equation \(\dfrac{{\it dA}}{dt}=kA.\) If 25% of the substance disappears in 10 years, what is its half-life?

Question 53.

Population Growth The population of a suburb grows at a rate proportional to the population. Suppose the population doubles in size from \(4000\) to \(8000\) in an \(18\)-month period and continues at the current rate of growth.

1. Write a differential equation that models the population \(P\) at time \(t\) in months.
2. Find the general solution to the differential equation.
3. Find the particular solution to the differential equation with the initial condition \(P(0) =4000.\)
4. What will the population be in \(4\) years [\(t=48]\)?

Question 54.

Uninhibited Growth At any time \(t\) in hours, the rate of increase in the area, in millimeters squared (mm²), of a culture of bacteria is twice the area \(A\) of the culture.

1. Write a differential equation that models the area of the culture at time \(t.\)
2. Find the general solution to the differential equation.
3. Find the particular solution to the differential equation if \(A=\) \(10 \text{mm}^{2},\) when \(t=0\).

Question 55.

Radioactive Decay The amount \(A\) of the radioactive element radium in a sample decays at a rate proportional to the amount of radium present. The half-life of radium is \(1690\) years.

1. Write a differential equation that models the amount \(A\) of radium present at time \(t\).
2. Find the general solution to the differential equation.
3. Find the particular solution to the differential equation with the initial condition \(A(0) =8g.\)
4. How much radium will be present in the sample in \(100\) years?

Question 56.

Radioactive Decay Carbon-14 is a radioactive element present in living organisms. After an organism dies, the amount \(A\) of carbon-14 present begins to decline at a rate proportional to the amount present at the time of death. The half-life of carbon-14 is \(5730\) years.

1. Write a differential equation that models the rate of decay of carbon-14.
2. Find the general solution to the differential equation.
3. A piece of fossilized charcoal is found that contains \(30\%\) of the carbon-14 that was present when the tree it came from died. How long ago did the tree die?

Question 57.

World Population Growth  Barring disasters (human-made or natural), the population \(P\) of humans grows at a rate proportional to itself current size. According to the U.N. World Population studies, from 2005 to 2010 the population of the more developed regions of the world (Europe, North America, Australia, New Zealand, and Japan) grew at an annual rate of \(0.409\%\) per year.

1. Write a differential equation that models the growth rate of the population.
2. Find the general solution to the differential equation.
3. Find the particular solution to the differential equation if in \(2010\) (\(t=0\)), the population of the more developed regions of the world was \(1.2359\times 10^{9}\).
4. If the rate of growth continues to follow this model, what is the projected population of the more developed regions in \(2020\)?

Question 58.

National Population Growth  Barring disasters (human-made or natural), the population \(P\) of humans grows at a rate proportional to its current size. According to the U.N. World Population studies, from 2005 to 2010 the population of Ecuador grew at an annual rate of \(1.490\%\) per year. Assuming this growth rate continues:

1. Write a differential equation that models the growth rate of the population.
2. Find the general solution to the differential equation.
3. Find the particular solution to the differential equation if in 2010 (\(t=0\)), the population of Ecuador was \(1.4465\times 10^{7}.\)
4. If the rate of growth continues to follow this model, when will the projected population of Ecuador reach \(20\) million persons?

Question 59.

Oetzi the Iceman was found in 1991 by a German couple who were hiking in the Alps near the border of Austria and Italy. Carbon-14 testing determined that Oetzi died 5300 years ago. Assuming the half-life of carbon-14 is 5730 years, what percent of carbon-14 was left in his body? (An interesting note: In September 2010 the complete genome mapping of Oetzi was completed.)

Question 60.

Uninhibited Decay Radioactive beryllium is sometimes used to date fossils found in deep-sea sediments. (Carbon-14 dating cannot be used for fossils that lived underwater.) The decay of beryllium satisfies the equation \(\dfrac{{\it dA}}{dt}=-\alpha A\), where \(\alpha =1.5\times 10^{-7}\) and \(t\) is measured in years. What is the half-life of beryllium?

Question 61.

Decomposition of Sucrose  Reacting with water in an acidic solution at \(35{}^{\circ}{\rm C}\), sucrose (C\(_{12}\)H\(_{22}\)O\(_{11})\) decomposes into glucose (C\(_{6}\)H\(_{12}\)O\(_{6}\)) and fructose (C\(_{6}\)H\(_{12}\)O\(_{6}\)) according to the law of uninhibited decay. An initial amount of 0.4mol of sucrose decomposes to 0.36 mol in \(30\) min. How much sucrose will remain after 2h? How long will it take until 0.10 mol of sucrose remains?

Question 62.

Chemical Dissociation  Salt (NaCl) dissociates in water into sodium (Na\(^{+}\)) and chloride (Cl\(^{-}\)) ions at a rate proportional to its mass. The initial amount of salt is 25 kg, and after 10 h, 15 kg are left.

1. How much salt will be left after 1 day?
2. After how many hours will there be less than \(\dfrac{1}{2}\) kg of salt left?

Question 63.

Voltage Drop The voltage of a certain condenser decreases at a rate proportional to the voltage. If the initial voltage is 20, and 2 s later it is 10, what is the voltage at time \(t\)? When will the voltage be 5?

Question 64.

Uninhibited Growth  The rate of change in the number of bacteria in a culture is proportional to the number present. In a certain laboratory experiment, a culture has 10,000 bacteria initially, 20,000 bacteria at time \(t_{1}\) minutes, and 100,000 bacteria at \((t_{1}+10)\) minutes.

1. In terms of \(t\) only, find the number of bacteria in the culture at any time \(t\) minutes (\(t\geq 0\)).
2. How many bacteria are there after \(20\) min?
3. At what time are 20,000 bacteria observed? That is, find the value of \(t_{1}\).

Question 65.

Verify that \(\int x\sqrt{x}\,dx\,\neq \,\big(\int {x\,dx}\big) \big(\int \sqrt{x}\,dx\big) \).

Question 66.

Verify that \(\int x(x^{2} + 1)\,dx\,\neq \,x\int\, (x^{2}+1)\,dx \).

Question 67.

Verify that \(\int \dfrac{{x^{2} - 1}}{x - 1}\,dx\,\neq \dfrac{\int ( x^{2}-1) dx}{\int (x-1) ~dx}\).

Question 68.

Prove that \(\int\, [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx.\)

Question 69.

Derive the integration formula \(\int a^{x}\,dx=\dfrac{a^{x}}{\ln a}+C\), \(a>0,\) \(a\neq 1.\) (Hint: Begin with the derivative of \(y=a^{x} \).)

Question 70.

Use the formula from Problem 69 to find:

1. \(\int 2^{x}dx\)
2. \(\int 3^{x}dx\)

Question 71.

1. Find \(y' \) if \(y=\ln \left\vert \tan \left(\dfrac{x}{2}+\dfrac{\pi }{4}\right) \right\vert \).
2. Use the result to show that \[ \int \sec x\,dx=\ln \left\vert \tan \left( \dfrac{x}{2}+\dfrac{\pi }{4}\right) \right\vert +\, C \]
3. Show that \(\ln \left\vert \tan \left( \dfrac{x}{2}+\dfrac{\pi }{4}\right) \right\vert =\ln \left\vert \sec x+\tan x\right\vert \).

Question 72.

1. Find \(y^{\prime }\) if \(y = x \sin^{-1}x+\sqrt{1-x^{2}}\).
2. Use the result to show that \[ \int \sin^{-1}x\,dx=x\sin ^{-1}x + \sqrt{1-x^{2}}+C \]

Question 73.

1. Find \(y^{\prime }\) if \(y=\dfrac{1}{2}x\sqrt{a^{2}-x^{2}}+\dfrac{1}{2}a^{2}\sin ^{-1}\left( \dfrac{x}{a}\right)\).
2. Use the result to show that \[ \int \sqrt{a^{2}-x^{2}} \,dx=\dfrac{1}{2}x\sqrt{a^{2}-x^{2}} +\,\dfrac{1}{2}a^{2}\sin ^{-1}\left( \dfrac{x}{a}\right) +C \]

Question 74.

1. Find \(y^{\prime }\) if \(y=\ln \left\vert \csc x-\cot x\right\vert\).
2. Use the result to show that \[ \int \csc x\,dx=\ln \left\vert \csc x-\cot x\right\vert + C \]

Question 75.

Gudermannian Function

1. Graph \(y=\,\)gd\((x)=\tan ^{-1}(\sinh x)\). This function is called the gudermannian of \(x\) (named after Christoph Gudermann).
2. If \(y=\,\)gd\((x)\), show that \(\cos y= \hbox{sech }x\) and \(\sin y=\tanh x\).
3. Show that if \(y=\,\)gd\((x)\), then \(y\) satisfies the differential equation \(y^{\prime }=\cos y\).
4. Use the differential equation of (c) to obtain the formula \[ \int \sec y \,d y = {\it gd}^{-1\,} (y) +C \] Compare this to \(\int \sec x\,dx\) \( =\, \ln \left\vert \sec x+\tan x\right\vert +C\).

Question 76.

The formula \(\dfrac{d}{dx}\int f(x)\,dx=f(x)\) says that if a function is integrated and the result is differentiated, the original function is returned. What about the other way around? Is the formula \(\int {f^\prime (x) dx=f(x)}\) correct? Be sure to justify your answer.