Question 1.

If the substitution \(u=2x+3\) is used with \(\int \sin (2x+3)\, dx,\) the result is \(\int\) _________\(du.\)

Question 2.

True or False  If the substitution \(u=x^{2}+3\) is used with \(\int_{0}^{1}x(x^{2}+3)^{3}dx,\) the result is \(\int_{0}^{1}x(x^{2}+3) ^{3}dx=\dfrac{1}{2}\int_{0}^{1}u^{3}du.\)

Question 3.

Multiple Choice  \(\int_{-4}^{4}x^{3}dx= [{\boldsymbol (a)}\, 128\enspace {\boldsymbol (b)}\, 4\enspace {\boldsymbol (c)}\,0\enspace {\boldsymbol (d)}\,64 ]\).

Question 4.

True or False \(\int_{0}^{5}x^{2}dx=\dfrac{1}{2}\int_{-5}^{5}x^{2}dx\).

Question 5.

\(\int e^{3x+1}dx\); let \(u=3x+1\).

Question 6.

\(\int \dfrac{dx}{x\ln x}\); let \(u=\ln x\).

Question 7.

\(\int {({1-t^{2}})^{6}t\,dt};\) let \(u=1-t^{2} \).

Question 8.

\(\int {\sin ^{5}x\cos x\,dx;}\) let \(u=\sin x \).

Question 9.

\(\int \dfrac{x^{2}\,dx}{\sqrt{1-x^{6}}}\ ;\) let \(u=x^{3}\).

Question 10.

\(\int \dfrac{e^{-x}}{6+e^{-x}} dx;\) let \(u=6+e^{-x}\).

Question 11.

\(\int \sin (3x) \,dx\)

Question 12.

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

Question 13.

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

Question 14.

\(\int \tan ^{2}x\sec ^{2}x\,dx \)

Question 15.

\(\int \dfrac{e^{1/x}}{x^{2}} dx\)

Question 16.

\(\int \dfrac{e^{\sqrt[3]{x}}}{\sqrt[3]{x^{2}}} dx\)

Question 17.

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

Question 18.

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

Question 19.

\(\int \dfrac{e^{x}}{\sqrt{1+e^{x}}} dx\)

Question 20.

\(\int \dfrac{dx}{x(\ln x)^{7}}\)

Question 21.

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

Question 22.

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

Question 23.

\(\int \dfrac{3e^{x}}{\sqrt[4]{e^{x}-1}} dx\)

Question 24.

\(\int \dfrac{[ \ln ( 5x) ] ^{3}}{x} dx\)

Question 25.

\(\int \dfrac{\cos x\;dx}{2\sin x-1}\)

Question 26.

\(\int \dfrac{\cos (2x)\;dx}{\sin (2x)} \)

Question 27.

\(\int \sec (5x) \,dx\)

Question 28.

\(\int \tan (2x) \,dx\)

Question 29.

\(\int \sqrt{\tan x}\sec^{2}x\,dx\)

Question 30.

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

Question 31.

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

Question 32.

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

Question 33.

\(\int \sin x\cdot e^{\cos x}\,dx\)

Question 34.

\(\int \sec ^{2}x\cdot e^{\tan x}\,dx\)

Question 35.

\(\int {x\sqrt{x+3}\,dx}\)

Question 36.

\(\int {x\sqrt{4-x}\,dx}\)

Question 37.

\(\int\, [ \sin x+\cos (3x) ] dx\)

Question 38.

\(\int\, \Big[ x^{2}+\sqrt{3x+2}\Big] dx\)

Question 39.

\(\int \dfrac{dx}{x^{2}+25}\)

Question 40.

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

Question 41.

\(\int \dfrac{dx}{\sqrt{9-x^{2}}}\)

Question 42.

\(\int \dfrac{dx}{\sqrt{16-9x^{2}}} \)

Question 43.

\(\int \sinh x\cosh x\,dx\)

Question 44.

\(\int \hbox{ sech}^{2}x\tanh x\,dx \)

Question 45.

\(\int_{-2}^{0}\dfrac{x}{(x^{2}+3)^{2}} dx\)

Question 46.

\(\int_{-1}^{1}( s^{2}-1) ^{5}s ds\)

Question 47.

\(\int_{0}^{1}x^{2}e^{x^{3}+1} dx\)

Question 48.

\(\int_{0}^{1}xe^{x^{2}-2} dx\)

Question 49.

\(\int_{1}^{6}x\sqrt{x+3}\,dx \)

Question 50.

\(\int_{2}^{6}x^{2}\sqrt{x-2} dx\)

Question 51.

\(\int_{0}^{2}x\cdot 3^{2x^{2}} dx\)

Question 52.

\(\int_{0}^{1}x\cdot 10^{-x^{2}} dx\)

Question 53.

\(\int_{1}^{3}\dfrac{1}{x^{2}}\sqrt{1-\dfrac{1}{x}} dx\)

Question 54.

\(\int_{0}^{\pi /4}\dfrac{\sin (2x)}{\sqrt{5-2\cos (2x) }} dx\)

Question 55.

\(\int_{0}^{2}\dfrac{e^{2x}}{e^{2x}+1} dx \)

Question 56.

\(\int_{1}^{3}\dfrac{e^{3x}}{e^{3x}-1} dx\)

Question 57.

\(\int_{2}^{3}\dfrac{dx}{x\ln x}\)

Question 58.

\(\int_{2}^{3}\dfrac{dx}{x(\ln x)^{2}}\)

Question 59.

\(\int_{0}^{\pi }e^{x}\cos (e^{x}) \,dx\)

Question 60.

\(\int_{0}^{\pi }e^{-x}\cos (e^{-x}) \,dx\)

Question 61.

\(\int_{0}^{1}\dfrac{x\,dx}{1+x^{4}}\)

Question 62.

\(\int_{0}^{1}\dfrac{e^{x}}{1+e^{2x}} dx\)

Question 63.

\(\int_{-2}^{2}( x^{2}-4) dx\)

Question 64.

\(\int_{-1}^{1}( x^{3}-2x) dx\)

397

Question 65.

\(\int_{-\pi /2}^{\pi /2}\dfrac{1}{3}\sin \theta d\theta \)

Question 66.

\(\int_{-\pi /4}^{\pi /4}\sec^{2}x\,dx\)

Question 67.

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

Question 68.

\({{\int_{-5}^{5}}} \big(x^{1/3}+x\big){ dx}\)

Question 69.

\(\int_{-5}^{5}\vert 2x\vert \,dx\)

Question 70.

\(\int_{-1}^{1}[\vert x\vert -3 ] \,dx\)

Question 71.

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

Question 72.

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

Question 73.

\(\int \left({2\sqrt{x^{2}+3}-{\dfrac{{4}}{{x}}}+9}\right) ^{6}\left({{\dfrac{{x}}{\sqrt{x^{2}+3}}}+{\dfrac{{2}}{{x^{2}}}}}\right) dx\)

Question 74.

\(\int {{\Big[\sqrt{({z^{2}+1})^{4}-3}\Big]}{\Big[z{({z^{2}+1})^{3}}\Big]}\,dz}\)

Question 75.

\(\int {{\dfrac{{x+4x^{3}}}{\sqrt{x}}}\,dx}\)

Question 76.

\(\int {{\dfrac{{z\,dz}}{{z+\sqrt{z^{2}+4}}}}}\)

Question 77.

\(\int {\sqrt{t}\sqrt{4+t\sqrt{t}}\,dt } \)

Question 78.

\(\int_{0}^{1}\dfrac{x+1}{x^{2}+3}dx\)

Question 79.

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

Question 80.

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

Question 81.

\(\int \dfrac{\sin x}{\sqrt{4-\cos ^{2}x}}dx\)

Question 82.

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

Question 83.

\(\int_{0}^{1}\dfrac{(z^{2}+5)(z^{3}+15z-3)\,}{{196-(z^{3}+15z-3)^{2}}}dz\)

Question 84.

\(\int_{2}^{17}\dfrac{dx}{\sqrt{\sqrt{{x-1}}+(x-1)^{5/4}}}\)

Question 85.

\(\int \dfrac{dx}{x\log _{10}x}\)

Question 86.

\(\int \dfrac{dx}{x\log _{3}\sqrt[5]{x}}\)

Question 87.

\(\int_{10}^{100}\dfrac{dx}{x\log x}\)

Question 88.

\(\int_{3}^{32}\dfrac{dx}{x\log_2 x}\)

Question 89.

\(\int_{3}^{9}\dfrac{dx}{x\log _{3}x} \)

Question 90.

\(\int_{10}^{100}\dfrac{dx}{x\log _{5}x} \)

Question 91.

If \(\int_{1}^{b}t^{2}(5t^{3}-1)^{1/2}\,dt=\dfrac{38}{45},\) find \(b.\)

Question 92.

If \(\int_{a}^{3}t\sqrt{9-t^{2}}\,dt=6,\) find \(a.\)

Question 93.

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

Question 94.

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

Question 95.

\(\int_{-\pi /2}^{\pi /2}\,\cos (2x+\pi )\,dx \)

Question 96.

\(\int_{{-}\pi /4}^{\pi /4}\sin ( 7\theta -\pi)\,d\theta \)

Question 97.

Area  Find the area under the graph of \(f(x) =\dfrac{x^{2}}{\sqrt{2x+1}}\) from \(0\) to \(4.\)

Question 98.

Area  Find the area under the graph of \(f(x) =\dfrac{x}{(x^{2}+1) ^{2}}\) from \(0\) to \(2.\)

Question 99.

Area  Find the area under the graph of \(y=\dfrac{1}{3x^{2}+1}\) from \(x=0\) to \(x=1\).

Question 100.

Area  Find the area under the graph of \(y=\dfrac{1}{x\sqrt{x^{2}-4}}\) from \(x=3\) to \(x=4\).

Question 101.

Area  Find the area under the graph of the catenary, \[ y=a\cosh \dfrac{x}{a}+b-a, \]

from \(x=0\) to \(x=a\).

Question 102.

Area  Find \(b\) so that the area under the graph of \[ y=(x+1)\sqrt{x^{2}+2x+4} \]

is \(\dfrac{56}{3}\) for \(0\leq x\leq b\).

Question 103.

Average Value  Find the average value of \(y=\tan x\) on the interval \(\left[ 0,\dfrac{\pi }{4}\right] \).

Question 104.

Average Value  Find the average value of \(y=\sec x\) on the interval \(\left[ 0,\dfrac{\pi}{4}\right] \).

Question 105.

If \(\int_{0}^{2}f(x - 3)\,dx = 8\), find \({\int_{-3}^{-1}{f(x)\,dx. }} \)

Question 106.

If \(\int_{-2}^{1}f(x + 1)\,dx=\dfrac{5}{2}\), find \({\int_{-1}^{2}{f(x)\,dx.}}\)

Question 107.

If \(\int_{0}^{4}f\left( \dfrac{x}{2}\right) dx = 8\), find \({\int_{0}^{2}{f(x)\,dx.}}\)

Question 108.

If \(\int_{0}^{1}g(3x)\,dx = 6\), find \({\int_{0}^{3}}g{(x)\,dx.}\)

Question 109.

Newton's Law of Cooling  Newton's Law of Cooling states that the rate of change of temperature with respect to time is proportional to the difference between the temperature of the object and the ambient temperature. A thermometer that reads \(4{}^{\circ}{\rm C}\) is brought into a room that is \(30{}^{\circ}{\rm C}\).

1. Write the differential equation that models the temperature \(u=u(t) \) of the thermometer at time \(t\) in minutes (min).
2. Find the general solution of the differential equation.
3. If the thermometer reads \(10{}^{\circ}{\rm C}\) after 2 min, determine the temperature reading 5 min after the thermometer is first brought into the room.

Question 110.

Newton's Law of Cooling  A thermometer reading \(70{}^{\circ}{\rm F}\) is taken outside where the ambient temperature is \(22{}^{\circ}{\rm F}\). Four minutes later the reading is \(32{}^{\circ}{\rm F}\).

1. Write the differential equation that models the temperature \(u=u(t) \) of the thermometer at time \(t.\)
2. Find the general solution of the differential equation.
3. Find the particular solution to the differential equation, using the initial condition that when \(t = 0\) min, then \(u=70{}^{\circ}{\rm F}.\)
4. Find the thermometer reading 7 min after the thermometer was brought outside.
5. Find the time it takes for the reading to change from \(70{}^{\circ}{\rm F}\) to within \(\dfrac{1}{2}{}^{\circ}{\rm F}\) of the air temperature.

Question 111.

Forensic Science  At \(4\) p.m., a body was found floating in water whose temperature is \(12{}^{\circ}{\rm C}.\) When the woman was alive, her body temperature was \(37{}^{\circ}{\rm C}\) and now it is \(20{}^{\circ}{\rm C}.\) Suppose the rate of change of the temperature \(u =u(t)\) of the body with respect to the time \(t\) in hours (h) is proportional to \(u(t)-T\), where \(T\) is the water temperature and the constant of proportionality is \(-0.159.\)

1. Write a differential equation that models the temperature \(u=u(t) \) of the body at time \(t\).
2. Find the general solution of the differential equation.
3. Find the particular solution to the differential equation, using the initial condition that at the time of death, when \(t=0 h\), her body temperature was \(u=37{}^{\circ}{\rm C}.\)
4. At what time did the woman drown?
5. How long does it take for the woman's body to cool to 15\(^{\circ}\)C?

Question 112.

Newton's Law of Cooling  A pie is removed from a \(350{}^{\circ}{\rm F}\) oven to cool in a room whose temperature is \(72{}^{\circ}{\rm F}\).

1. Write the differential equation that models the temperature \(u=u(t) \) of the pie at time \(t.\)
2. Find the general solution of the differential equation.
3. Find the particular solution to the differential equation, using the initial condition that when \(t=0 min\), then \(u=350{}^{\circ}{\rm F}.\)
4. If \(u(5) =\) \(200{}^{\circ}{\rm F},\) what is the temperature of the pie after \(15\) min?
5. How long will it take for the pie to be \(100{}^{\circ}{\rm F}\) and ready to eat?

Question 113.

Electric Potential  The electric field strength a distance \(z \) from the axis of a ring of radius \(R\) carrying a charge \(Q\) is given by the formula \[ E(z)={\dfrac{{Qz}}{{(R^{2}+z^{2})^{3/2}}}} \]

If the electric potential \(V\) is related to \(E\) by \(E=-\dfrac{dV}{dz}\), what is \(V(z)\)?

Question 114.

Impulse During a Rocket Launch  The impulse \(J\) due to a force \(F\) is the product of the force times the amount of time \(t\) for which the force acts. When the force varies over time, \[ J=\int_{t_{1}}^{t_{2}}F(t)\,dt. \]

We can model the force acting on a rocket during launch by an exponential function \(F(t)=Ae^{bt}\), where \(A\) and \(b\) are constants that depend on the characteristics of the engine. At the instant lift-off occurs (\(t=0\)), the force must equal the weight of the rocket.

1. Suppose the rocket weighs 25,000 N (a mass of about 2500 kg or a weight of 5500lb), and 30 seconds after lift-off the force acting on the rocket equals twice the weight of the rocket. Find \(A\) and \(b.\)
2. Find the impulse delivered to the rocket during the first 30 seconds after the launch.

Question 115.

Air Resistance on a Falling Object  If an object of mass \(m\) is dropped, the air resistance on it when it has speed \(v\) can be modeled as \(F_{{\rm air}}=-kv\), where the constant \(k\) depends on the shape of the object and the condition of the air. The minus sign is necessary because the direction of the force is opposite to the velocity. Using Newton's Second Law of Motion, this force leads to a downward acceleration \(a(t)=ge^{-kt/m} \). See Problem 137. Using the equation for \(a(t)\), find:

1. \(v(t)\), if the object starts from rest \(v_0=v(0)=0\), with the positive direction downward.
2. \(s(t)\), if the object starts from the position \(s_0=s(0)=0\), with the positive direction downward.
3. What limits do \(a(t)\), \(v(t)\), and \(s(t)\) approach if the object falls for a very long time (\(t\rightarrow \infty \))? Interpret each result and explain if it is physically reasonable.
4. Graph \(a=a(t)\), \(v=v(t)\), \(s=s(t)\). Do the graphs support the conclusions obtained in part (c)? Use \(g = 9.8\) m\(/\)s\(^2\), \(k = 5\), and \(m = 10\) kg.

Question 116.

Area  Let \(f(x)=k\sin ( kx) \), where \(k\) is a positive constant.

1. Find the area of the region under one arch of the graph of \(f.\)
2. Find the area of the triangle formed by the \(x\)-axis and the tangent lines to one arch of \(f\) at the points where the graph of \(f\) crosses the \(x\)-axis.

Question 117.

Use an appropriate substitution to show that \[ \int_{0}^{1}x^{m}(1 - x)^{n} dx = \int_{0}^{1}x^{n}(1 - x)^{m} dx, \]

where \(m,\) \(n\) are positive integers.

Question 118.

Properties of Integrals  Find \({\int_{-1}^{1}{f(x)\,dx}}\) for the function given below: \[ f(x)={\left\{ {{ \begin{array}{l@{ }l@{ }l} {x+1} & \hbox{if} & {x\lt0} \\ \cos (\pi x) & \hbox{if} & {x\geq 0} \end{array} }}\right. } \]

Question 119.

If \(f\) is continuous on \([a,b]\), show that \[ \int_{a}^{b}f(x)\,dx = \int_{a}^{b}f(a + b -\,x)\,dx \]

Question 120.

If \(\int_{0}^{1}f(x) \,dx = 2\), find:

1. \({\int_{0}^{0.5}{f(2x)\,dx }} \)
2. \(\int_{0}^{3}f \left( \dfrac{1}{3}x\right) dx\)
3. \(\int_{0}^{1/5}f( 5x) dx\)
4. Find the upper and lower limits of integration so that \[ \int_{a}^{b}f\left( \dfrac{x}{4}\right) dx=8. \]
5. Generalize (d) so that \(\int_{a}^{b}f( kx) dx=\dfrac{1}{k}\cdot 2\) for \(k>0.\)

Question 121.

If \(\int_{0}^{2}f (s) \,ds = 5\), find:

1. \({\int_{-1}^{1}{f(s+1)\,ds }}\)
2. \(\int_{-3}^{-1}f( s+3) \,ds\)
3. \(\int_{4}^{6}f( s-4) \,ds\)
4. Find the upper and lower limits of integration so that \[ \int_{a}^{b}f(s - 2) ds=5. \]
5. Generalize (d) so that \(\int_{a}^{b}f( s-k) \,ds=5\) for \(k>0.\)

Question 122.

Find \(\int_{0}^{b}\vert 2x\vert \,dx\) for any real number \(b.\)

Question 123.

If \(f\) is an odd function, show that \(\int_{-a}^{a}f(x)\,dx = 0\).

Question 124.

Find the constant \(k\), where \(0\leq k\leq \,3\), for which \[ {\int_{0}^{3}{{\dfrac{{x}}{\sqrt{x^{2}+16}}}dx={\dfrac{{3k}}{\sqrt{k^{2}+16}}} }} \]

Question 125.

If \(n\) is a positive integer, for what number is \[ \int_{0}^{a}x^{n-1} dx= \dfrac{1}{n} \]

Question 126.

If \(f\) is a continuous function defined on the interval \([0,1]\), show that \[ {\int_{0}^{\pi }{x f(\sin x)\,dx={\dfrac{{\pi }}{{2}}}{\int_{0}^{\pi }{f(\sin x)\,dx}}}} \]

Question 127.

Prove that \(\int \csc x\,dx=\ln \left\vert \csc x-\cot x\right\vert +C\). [Hint: Multiply and divide the integrand by \((\csc x-\cot x)\).]

Question 128.

Describe a method for finding \(\int_{a}^{b}\vert f(x)\vert \,dx\) in terms of \(F(x)=\int f(x)\mathit{~}dx\) when \(f(x)\) has finitely many zeros.

Question 129.

Find \(\int \sqrt[n]{{a\,{+}\,bx}}\,dx\), where \(a\) and \(b\) are real numbers, \(b\neq 0\), and \(n\geq 2\) is an integer.

Question 130.

If \(f\) is continuous for all \(x\), which of the following integrals have the same value?

1. \(\int_{a}^{b}{f(x)\,dx }\)
2. \(\int_{0}^{b-a}{f(x+a)\,dx } \)
3. \(\int_{a+c}^{b+c}{f(x+c)\,dx }\)

Question 131.

Find \(\int {{\dfrac{{x^{6}+3x^{4}+3x^{2}+x+1}}{{(x^{2}+1)^{2}}}}}\,dx\).

Question 132.

Find \(\int \dfrac{\sqrt[4]{x}}{\sqrt{x}+\sqrt[3]{x}}dx\).

Question 133.

Find \(\int \dfrac{3x+2}{x\sqrt{x+1}}\ dx\).

Question 134.

Find \(\int \dfrac{dx}{(x\ln x) [\ln (\ln x)] }\).

Question 135.

Air Resistance on a Falling Object  (Refer to Problem 115.) If an object of mass \(m\) is dropped, the air resistance on it when it has speed \(v\) can be modeled as \[ F_{{\rm air}}=-kv, \]

where the constant \(k\) depends on the shape of the object and the condition of the air. The minus sign is necessary because the direction of the force is opposite to the velocity. Using Newton's Second Law of Motion, show that the downward acceleration of the object is \[ a(t) =ge^{-kt/m}, \]

where \(g\) is the acceleration due to gravity. (Hint: The velocity of the object obeys the differential equation \[ m\dfrac{dv}{dt}=mg-kv \]

Solve the differential equation for \(v\) and use the fact that \(ma=mg-kv\).)

Question 136.

A separable differential equation can be written in the form \(\dfrac{dy}{dx}=\dfrac{f(x)}{g (y)}\), where \(f\) and \(g\) are continuous. Then \[ \int g(y)\,dy=\int f(x)\,dx \]

and integrating (if possible) will give a solution to the differential equation. Use this technique to solve parts (a)–(c) below. (You may need to leave your answer in implicit form.)

1. \(\dfrac{y^{2}}{x}\dfrac{dy}{dx}=1+x^{2} \)
2. \(\dfrac{dy}{dx}=y\dfrac{x^{2}-2x+1}{y+3}\)
3. \(y\dfrac{dy}{dx}=\dfrac{x^{2}}{y+4}\);  if \(y=2\) when \(x=8\)