Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

Chapter 8

Integration Techniques, L’Hopital’s Rule, and Improper Integrals - all with Video Answers

Educators

Section 1

Basic Integration Rules

Problem 1

Select the correct antiderivative.
$$\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}$$
(a) $2 \sqrt{x^{2}+1}+C$
(b) $\sqrt{x^{2}+1}+C$
(c) $\frac{1}{2} \sqrt{x^{2}+1}+C$
(d) $\ln \left(x^{2}+1\right)+C$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 2

Select the correct antiderivative.
$$\frac{d y}{d x}=\frac{x}{x^{2}+1}$$
(a) $\ln \sqrt{x^{2}+1}+C$
(b) $\frac{2 x}{\left(x^{2}+1\right)^{2}}+C$
(c) arctan $x+C$
(d) $\ln \left(x^{2}+1\right)+C$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 3

Select the correct antiderivative.
$$\frac{d y}{d x}-\frac{1}{x^{2}+1}$$
(a) $\ln \sqrt{x^{2}+1}+C$
(b) $\frac{2 x}{\left(x^{2}+1\right)^{2}}+c$
(c) arctan $x+C$
(d) $\ln \left(x^{2}+1\right)+C$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 4

Select the correct antiderivative.
$$\frac{d y}{d x}=x \cos \left(x^{2}+1\right)$$
(a) $2 x \sin \left(x^{2}+1\right)+C$
(b) $-\frac{1}{2} \sin \left(x^{2}+1\right)+C$
(c) $\frac{1}{2} \sin \left(x^{2}+1\right)+C$
(d) $-2 x \sin \left(x^{2}+1\right)+C$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 5

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int(5 x-3)^{4} d x$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 6

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \frac{2 t+1}{t^{2}+t-4} d t$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 7

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \frac{1}{\sqrt{x}(1-2 \sqrt{x})} d x$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 8

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \frac{2}{(2 t-1)^{2}+4} d t$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 9

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \frac{3}{\sqrt{1-t^{2}}} d t$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 10

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \frac{-2 x}{\sqrt{x^{2}-4}} d x$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 11

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int t \sin t^{2} d t$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 12

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \sec 5 x \tan 5 x \, d x$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 13

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int(\cos x) e^{\sin x} d x$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 14

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \frac{1}{x \sqrt{x^{2}-4}} d x$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 15

Find the indefinite integral.
$$\int 14(x-5)^{6} d x$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 16

Find the indefinite integral.
$$\int \frac{5}{(t+6)^{3}} d t$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 17

Find the indefinite integral.
$$\int \frac{7}{(z-10)^{7}} d z$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 18

Find the indefinite integral.
$$\int t^{3} \sqrt{t^{4}+1} d t$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 19

Find the indefinite integral.
$$\int\left[v+\frac{1}{(3 v-1)^{3}}\right] d v$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 20

Find the indefinite integral.
$$\int\left[4 x-\frac{2}{(2 x+3)^{2}}\right] d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 21

Find the indefinite integral.
$$\int \frac{t^{2}-3}{-t^{3}+9 t+1} d t$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 22

Find the indefinite integral.
$$\int \frac{x+1}{\sqrt{3 x^{2}+6 x}} d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 23

Find the indefinite integral.
$$\int \frac{x^{2}}{x-1} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 24

Find the indefinite integral.
$$\int \frac{3 x}{x+4} d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 25

Find the indefinite integral.
$$\int \frac{e^{x}}{1+e^{x}} d x$$

Fuzail Shakir

Numerade Educator

Problem 26

Find the indefinite integral.
$$\int\left(\frac{1}{2 x+5}-\frac{1}{2 x-5}\right) d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 27

Find the indefinite integral.
$$\int\left(5+4 x^{2}\right)^{2} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 28

Find the indefinite integral.
$$\int x \cos 2 \pi x^{2} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 29

Find the indefinite integral.
$$\int \csc \pi x \cot \pi x \, d x$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 30

Find the indefinite integral.
$$\int \frac{\sin x}{\sqrt{\cos x}} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 31

Find the indefinite integral.
$$\int \csc ^{2} x e^{\cot x} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 32

Find the indefinite integral.
$$\int \frac{2}{e^{-x}+1} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 33

Find the indefinite integral.
$$\int \frac{2}{7 e^{x}+4} d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 34

Find the indefinite integral.
$$\int \frac{\ln x^{2}}{x} d x$$

Vikash Ranjan

Numerade Educator

Problem 35

Find the indefinite integral.
$$\int(\tan x)[\ln (\cos x)] d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 36

Find the indefinite integral.
$$\int \frac{1}{\cos \theta-1} d \theta$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 37

Find the indefinite integral.
$$\int \frac{1}{\cos \theta-1} d \theta$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 38

Find the indefinite integral.
$$\int \frac{1+\cos \alpha}{\sin \alpha} d \alpha$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 39

Find the indefinite integral.
$$\int \frac{-1}{\sqrt{1-(4 t+1)^{2}}} d t$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 40

Find the indefinite integral.
$$\int \frac{1}{25+4 x^{2}} d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 41

Find the indefinite integral.
$$\int \frac{e^{1 / 2}}{t^{2}} d t$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 42

Find the indefinite integral.
$$\int \frac{\tan (2 / t)}{t^{2}} d t$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 43

Find the indefinite integral.
$$\int \frac{6}{\sqrt{10 x-x^{2}}} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 44

Find the indefinite integral.
$$\int \frac{1}{(x-1) \sqrt{4 x^{2}-8 x+3}} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 45

Find the indefinite integral.
$$\int \frac{4}{4 x^{2}+4 x+65} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 46

Find the indefinite integral.
$$\int \frac{1}{x^{2}-4 x+9} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 47

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 MathGraphs.com.
$$\begin{aligned}&\frac{d s}{d t}=\frac{t}{\sqrt{1-t^{4}}}\\&\left(0,-\frac{1}{2}\right)\end{aligned}$$
(GRAPH CAN'T COPY)

Audrey Fong

Numerade Educator

Problem 48

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 MathGraphs.com.
$$\begin{aligned}&\frac{d y}{d x}=\frac{1}{\sqrt{4 x-x^{2}}}\\&\left(2, \frac{1}{2}\right)
\end{aligned}$$
(GRAPH CAN'T COPY)

Audrey Fong

Numerade Educator

Problem 49

Use a computer algebra system to graph the slope ficld for the differential equation and zraph the solution through the specified initial condition.
$$\frac{d y}{d x}=0.8 y, y(0)=4$$

Audrey Fong

Numerade Educator

Problem 50

Use a computer algebra system to graph the slope ficld for the differential equation and zraph the solution through the specified initial condition.
$$\frac{d y}{d x}=5-y, y(0)=1$$

Audrey Fong

Numerade Educator

Problem 51

Solve the differ. ential equation.
$$\frac{d y}{d x}=\left(e^{x}+5\right)^{2}$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 52

Solve the differ. ential equation.
$$\frac{d y}{d x}=\left(4-e^{2 x}\right)^{2}$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 53

Solve the differ. ential equation.
$$\frac{d r}{d t}=\frac{10 t}{\sqrt{1-e^{3 t}}}$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 54

Solve the differ. ential equation.
$$\frac{d r}{d t}=\frac{\left(1+e^{\eta^{2}}\right.}{e^{3 t}}$$

Audrey Fong

Numerade Educator

Problem 55

Solve the differ. ential equation.
$$\left(4+\tan ^{2} x\right) y^{\prime}=\sec ^{2} x$$

Ben Ault

Numerade Educator

Problem 56

Solve the differ. ential equation.
$$y^{\prime}=\frac{1}{x \sqrt{4 x^{2}-9}}$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 57

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{\pi / 4} \cos 2 \pi d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 58

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{\pi} \sin ^{2} t \cos t d t$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 59

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{1} x e^{-x^{3}} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 60

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{1}^{1} 1-\ln x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 61

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{\infty} \frac{2 x}{\sqrt{x^{2}+36}} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 62

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{1}^{3} \frac{2 x^{2}+3 x-2}{x} d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 63

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{2 / \sqrt{3}} \frac{1}{4+9 x^{2}} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 64

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{7} \frac{1}{\sqrt{100-x^{2}}} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 65

Find the area of the region.
$$y=(-4 x+6)^{3 / 2}$$
(GRAPH CAN'T COPY)

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 66

Find the area of the region.
$$y=\frac{3 x+2}{x^{2}+9}$$
(GRAPH CAN'T COPY)

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 67

Find the area of the region.
$$z=x^{2}\left(1-x^{2}\right)$$
(GRAPH CAN'T COPY)

Audrey Fong

Numerade Educator

Problem 68

Find the area of the region.
$$y=\sin 2 x$$
(GRAPH CAN'T COPY)

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 69

Use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two antiderivatives.
$$\int \frac{1}{x^{2}+4 x+13} d x$$

Audrey Fong

Numerade Educator

Problem 70

Use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two antiderivatives.
$$\int \frac{x-2}{x^{2}+4 x+13} d x$$

Audrey Fong

Numerade Educator

Problem 71

Use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two antiderivatives.
$$\int \frac{1}{1+\sin \theta} d \theta$$

Audrey Fong

Numerade Educator

Problem 72

Use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two antiderivatives.
$$\int\left(\frac{e^{x}+e^{-x}}{2}\right)^{3} d x$$

Audrey Fong

Numerade Educator

Problem 73

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int x\left(x^{2}+1\right)^{3} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 74

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int x \sec \left(x^{2}+1\right) \tan \left(x^{2}+1\right) d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 75

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int \frac{x}{x^{2}+1} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 76

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int \frac{1}{x^{2}+1} d x$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 77

Determine the constants $a$ and $b$ such that
$\sin x+\cos x=a \sin (x+b)$
Use this result to integrate
$\int \frac{d x}{\sin x+\cos x}$

Pravakar Paul

Numerade Educator

Problem 78

Show that $\sec x=\frac{\sin x}{\cos x}+\frac{\cos x}{1+\sin x}$
Then use this identity to derive the basic integration rule
$\int \sec x \, d x=\ln |\sec x+\tan x|+C$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 79

The graphs of $f(x)=x$ and $g(x)=a x^{2}$ intersect at the points (0,0) and $(1 / a, 1 / a) .$ Find $a(a>0)$ such that the area of the region bounded by the graphs of these two functions is $\frac{2}{3}$

Ashley Boni

Numerade Educator

Problem 80

When evaluating $\int_{-1}^{1} x^{2} d x$
is it appropriate to substitute $u=x^{2}, \quad x=\sqrt{u}, \quad$ and $\quad d x=\frac{d u}{2 \sqrt{u}}$ to obtain
$\frac{1}{2} \int_{1}^{1} \sqrt{u} d u=0 ?$ Explain.

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 81

(a) Explain why the antiderivative $y_{1}=e^{x+c_{1}}$ is equivalent to the antiderivative $y_{2}=C e^{x}$
(b) Explain why the antiderivative $y_{1}=\sec ^{2} x+C_{1}$ is equivalent to the antiderivative $y_{2}=\tan ^{2} x+C$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 82

Using the graph, is $\int_{0}^{5} f(x) d x$ positive or negative? Explain.
(GRAPH CAN'T COPY)

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 83

$f(x)=\frac{4 x}{x^{2}+1}, \quad[0,2]$
(a) 3
(b) 1
$(c)-8$
(d) 8
(e) 10

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 84

Determine which value best approximates the area of the region between the $x$ -axis and the function over the given interval. (Make your selection on the basis of a sketch of the region and not by integrating.)
$f(x)=\frac{4 x}{x^{2}+1}, \quad[0,2]$
(a) 3
(b) 1
$(c)-8$
(d) 8
(e) 10

Audrey Fong

Numerade Educator

Problem 85

(a) sketch the region whose area is given by the integral, (b) sketch the solid whose volume is given by the integral when the disk method is used, and (c) sketch the solid whose volume is given by the integral when the shell method is used. (There is more than one correct answer for each part.)
$$\int_{0}^{2} 2 \pi x^{2} d x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 86

(a) sketch the region whose area is given by the integral, (b) sketch the solid whose volume is given by the integral when the disk method is used, and (c) sketch the solid whose volume is given by the integral when the shell method is used. (There is more than one correct answer for each part.)
$$\int_{0}^{4} \pi y d y$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 87

The region bounded by $y=e^{-x^{2}}, y=0, x=0$ and $x=b(b>0)$ is revolved about the $y$ -axis.
(a) Find the volume of the solid generated when $b=1$
(b) Find $b$ such that the volume of the generated solid is $\frac{4}{3}$ cubic units.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 88

Consider the region bounded by the graphs of $x=0, y=\cos x^{2}, y=\sin x^{2},$ and $x=\sqrt{\pi / 2}$. Find the
volume of the solid generated by revolving the region about the $y$ -axis.

Taylor Francis

Numerade Educator

Problem 89

Find the arc length of the graph of $y=\ln (\sin x)$ from $x=\pi / 4$ to $x=\pi / 2$

Pravakar Paul

Numerade Educator

Problem 90

Find the arc length of the graph of $y=\ln (\cos x)$ from $x=0$ to $x=\pi / 3$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 91

Find the area of the surface formed by revolving the graph of $y=2 \sqrt{x}$ on the interval [0,9] about the $x$ -axis.

K B

Numerade Educator

Problem 92

Find the $x$ -coordinate of the centroid of the region bounded by the graphs of
$y=\frac{5}{\sqrt{25-x^{2}}}, \quad y=0, \quad x=0, \quad$ and $\quad x=4$

Dwijendra Rao

Numerade Educator

Problem 93

Find the average value of the function over the given interval.
$$f(x)=\frac{1}{1+x^{2}}, \quad-3 \leq x \leq 3$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 94

Find the average value of the function over the given interval.
$f(x)=\sin n x, \quad 0 \leq x \leq \pi / n, n$ is a positive integer.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 95

Use the integration
-capabilities of a graphing utility to approximate the are length
- of the curve over the given interval.
$$y=\tan \pi x, \quad\left[0, \frac{1}{4}\right]$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 96

Use the integration
-capabilities of a graphing utility to approximate the are length
- of the curve over the given interval.
$$y=x^{2 / 3}, \quad[1,8]$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 97

(a) Find $\int \cos ^{3} x d x$
(b) Find $\int \cos ^{5} x d x$
(c) Find $\int \cos ^{7} x d x$
(d) Explain how to find $\int \cos ^{15} x d x$ without actually integrating.

Pravakar Paul

Numerade Educator

Problem 98

(a) Write $\int \tan ^{3} x d x$ in terms of $\int \tan x \, d x$. Then find $\int \tan ^{3} x d x$
(b) Write $\int \tan ^{5} x d x$ in terms of $\int \tan ^{3} x d x$
(c) Write $\int \tan ^{2 k+1} x d x,$ where $k$ is a positive integer, in terms of $\int \tan ^{2} x-1 x d x$
(d) Explain how to find $\int \tan ^{15} x d x$ without actually integrating.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 99

Show that the following results are equivalent.
Integration by tables:
$\int \sqrt{x^{2}+1} d x=\frac{1}{2}(x \sqrt{x^{2}+1}+\ln |x+\sqrt{x^{2}+1}|)+C$
Integration by computer algebra system:
$\int \sqrt{x^{2}+1} d x=\frac{1}{2}[x \sqrt{x^{2}+1}+\operatorname{arcsinh}(x)]+C$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 100

Evaluate $$\int_{2}^{4} \frac{\sqrt{\ln (9-x)} d x}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator