Question 5.306

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

Question 5.307

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 5.308

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

Question 5.309

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

Question 5.310

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

Question 5.311

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

Question 5.312

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

Question 5.313

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

Question 5.314

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

Question 5.315

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

Question 5.316

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

Question 5.317

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

Question 5.318

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

Question 5.319

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

Question 5.320

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

Question 5.321

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

Question 5.322

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

Question 5.323

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

Question 5.324

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

Question 5.325

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

Question 5.326

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

Question 5.327

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

Question 5.328

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

Question 5.329

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

Question 5.330

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

Question 5.331

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

Question 5.332

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

Question 5.333

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

Question 5.334

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

Question 5.335

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

Question 5.336

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

Question 5.337

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

Question 5.338

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

Question 5.339

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

Question 5.340

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

Question 5.341

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

Question 5.342

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

Question 5.343

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

Question 5.344

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

Question 5.345

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

Question 5.346

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

Question 5.347

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

Question 5.348

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

Question 5.349

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

Question 5.350

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

Question 5.351

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

Question 5.352

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

Question 5.353

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

Question 5.354

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

Question 5.355

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

Question 5.356

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 5.357

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 5.358

Population Growth The population of a suburb grows at a rate proportional to the population. Suppose the population doubles insize 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 5.359

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 5.360

Radioactive Decay The amount \(A\) of the radioactive element radium in a sample decays at a rate proportional to the amount of radiumpresent. 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 5.361

Radioactive Decay Carbon-14 is a radioactive element present in living organisms. After an organism dies, the amount \(A\) ofcarbon-14 present begins to decline at a rate proportional to the amountpresent 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 5.362

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 5.363

National Population Growth  Barring disasters (human-made or natural), the population \(P\) of humans grows at a rate proportional to itscurrent 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 5.364

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 5.365

Uninhibited Decay Radioactive beryllium is sometimes used to date fossils found in deep-sea sediments. (Carbon-14 dating cannot beused for fossils that lived underwater.) The decay of beryllium satisfiesthe 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 5.366

Decomposition of Sucrose  Reacting with water in anacidic 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 ofuninhibited decay. An initial amount of 0.4mol of sucrosedecomposes to 0.36 mol in \(30\) min. How much sucrose will remainafter 2h? How long will it take until 0.10 mol of sucrose remains?

Question 5.367

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 5.368

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 5.369

Uninhibited Growth  The rate of change in the number ofbacteria in a culture is proportional to the number present. In a certainlaboratory experiment, a culture has 10,000 bacteria initially, 20,000bacteria 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 theculture 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 5.370

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

Question 5.371

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

Question 5.372

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

Question 5.373

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

Question 5.374

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 5.375

Use the formula from Problem 69 to find:

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

Question 5.376

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 5.377

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 5.378

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 5.379

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

Question 5.380

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 thedifferential 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 \(\int \sec x\,dx\) \( =\, \ln \left\vert \sec x+\tanx\right\vert +C\).

Question 5.381

The formula \(\dfrac{d}{dx}\int f(x)\,dx=f(x)\) says that if afunction 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.