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Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions
Section 5.1 The Natural Logarithmic Function: Differentiation
Objective: In this lesson you learned the properties of the natural logarithmic function and how to find the derivative of the natural logarithmic function.
Course
Number
Instructor
Date
Important Vocabulary Define each term or concept.
Natural logarithmic function The natural logarithmic function is defined by ln x =x∫ 1/t dt, x > 0. 1
e The letter e denotes the positive real number such thate
ln e = ∫ 1/t dt = 1.1
I. The Natural Logarithmic Function (Pages 324−326)
The domain of the natural logarithmic function is the set
of all positive real nu m bers .
The value of ln x is positive for x > 1 and negative
for 0 < x < 1 . Moreover, ln (1) = 0 ,
because the upper and lower limits of integration are equal
when x = 1 .
The natural logarithmic function has the following properties:
1. The domain is (0, ∞) and the range is (−∞, ∞).
2. The function is continuous, increasing, and one-to-one.
3. The graph is concave downward.
If a and b are positive numbers and n is rational, then the following properties are true:
1. ln (1) = 0 .
2. ln (ab) = ln a + ln b .
3. ln(an) = n ln a .
What you should learn How to develop and use properties of the natural logarithmic function
Larson/Edwards Calculus 9e Notetaking Guide IAECopyright © Cengage Learning. All rights reserved. 95
96 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions
Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights
b⎜4. ln ⎛ a ⎞ = l n a − ln b .
⎝ ⎠
Example 1: Expand the logarithmic expression ln
ln x + 4 ln y − ln 2
xy 4.
2
II. The Number e (Page 327)
The base for the natural logarithm is defined using the fact
that the natural logarithmic function is continuous, is one-to-
one, and has a range of (−∞, ∞). So, there must a unique real
number x such that ln x = 1 . This
number is denoted by the letter e ,
which has the decimal approximation
2. 71 828 1 82 8 46 .
III. The Derivative of the Natural Logarithmic Function(Pages 328−330)
Let u be a differentiable function of x. Complete the
following rules of differentiation for the natural logarithmic
function:d [ln x] = 1/x , x > 0dxd [ln u] = 1/u [du/dx] = u′ /u , u > 0dx
What you should learn How to understand the definition of the number e
What you should learn How to find derivatives of functions involving the natural logarithmic function
Example 2: Find the derivative ofx + 2x lnx
f (x) = x2 ln x .
If u is a differentiable function of x such that u ≠ 0 , thend
[ln u ] = u ′ / u . In other words, functions ofdxthe form
y = ln u
can be differentiated as if the absolu t e
value signs were not present .
Homework Assignment
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Page(s)
Exercises
∫
∫ ∫
∫
x
98 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions
Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights
Section 5.2 The Natural Logarithmic Function: Integration 97
Section 5.2 The Natural Logarithmic Function: Integration
Objective: In this lesson you learned how to find the antiderivative of the natural logarithmic function.
Course Number
Instructor
Date
I. Log Rule for Integration (Pages
334−337) Let u be a differentiable function of
x.
1 dx = ln| x | + C
x
u′ 1
u dx =
u du = ln| u | + C
What you should learn How to use the Log Rule for Integration to integrate a rational function
Example 1: Find ⎛1 − 1 ⎞ dx .
⎝ ⎠⎜ x ⎟
x − ln| x | + C
Example 2: Find2
∫ 3
− x3
dx .
− (1/3) ln|3 − x3| + C
Example 3: Find
∫x2 − 4x + 1
x
dx .
(1/2)x2 − 4x + ln| x | + C
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If a rational function has a numerator of degree greater than
or equal t o that of the d e no m inator ,
division may reveal a form to which you can apply the Log Rule.
9 Chapter Logarithmic, Exponential, and Other Transcendental
Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights
Guidelines for Integration
1. Learn a basic list of integration formulas.
2. Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula.
3. If you cannot find a u-substitution that works, try altering the integrand. You might try a trigonometric identity, multiplication and division by the same quantity, addition and subtraction of the same quantity, or long division. Be creative.
4. If you have access to computer software that will find antiderivatives symbolically, use it.
II. Integrals of Trigonometric Functions (Pages 338−339) What you should learn
How to integrate trigonometric functions
∫sin u du
=
- cos u + C
∫cos u du = sin u + C
∫ tan u du =
- ln|cos u | + C
∫cot u du = ln|sin u | + C
∫sec u du = ln | s ec u + tan u | + C
∫cscu du =
- l n |csc u + cot u | + C
Example 4: Find ∫csc5x dx
−(1/5)ln|csc 5x + cot 5x| + C
Homework Assignment
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Exercises
9Section Inverse
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Section 5.3 Inverse Functions
Objective: In this lesson you learned how to determine whether a function has an inverse function.
Course
Number
Instructor
Date
Important Vocabulary Define each term or concept.
Inverse function A function g is the inverse function of the function f if f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of f. The function g is denoted by f −1.Horizontal Line Test A test that states that a function f has an inverse function if and only if every horizontal line intersects the graph of f at most once.
I. Inverse Functions (Pages 343−344)
For a function f that is represented by a set of ordered pairs, you
can form the inverse function of f by interchang i ng the first
and second coordinates of each ordered pairs .
For a function f and its inverse f −1, the domain of f is equal
to the range of f −1 , and the range of f is equal to
the domain of f −1 .
State three important observations about inverse functions.
1. If g is the inverse function of f, then f is the inverse function of g.
2. The domain of f −1 is equal to the range of f, and the range off −1 is equal to the domain of f.
3. A function need not have an inverse function, but if it does, the inverse function is unique.
To verify that two functions, f and g, are inverse functions of
each other, . . . find f(g(x)) and g(f(x)). If both of these
compositions are equal to the identity function x for every x in
the domain of the inner function, then the functions are
inverses of each other.
What you should learn How to verify that one function is the inverse function of another function
1 Chapter Logarithmic, Exponential, and Other Transcendental
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Example 1: Verify that the functions f (x) = 2x − 3 and
g(x) = x +
3
2
are inverse functions of each other.
The graph of f −1 is a reflection of the graph of f in the line
y = x .
The Reflective Property of Inverse Functions states that the
graph of f contains the point (a, b) if and only if the
graph of f −1 contains the point (b, a) .
II. Existence of an Inverse Function (Pages
345−347) State two reasons why the horizontal line test
is valid.
1. A function has an inverse function if and only if it is one-to- one.
2. If f is strictly monotonic on its entire domain, then it is one- to-one and therefore has an inverse function.
Example 2: Does the graph of the function shown below have an inverse function? Explain.No, it doesn’t pass the Horizontal Line Test.
What you should learn How to determine whether a function has an inverse function
y5
3
-5 -3
1
x-1 1 3 5
-1
-3
-5
Complete the following guidelines for finding an inverse
function.
1Section Inverse
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1) Determine whether the function given by y = f(x) has an
inverse function.
1 Chapter Logarithmic, Exponential, and Other Transcendental
Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights
2) Solve for x as a function of y: x = g(y) = f −1(y).
3) Interchange x and y. The resulting equation is y = f −1(x).
4) Define the domain of f −1 to be the range of f.
5) Verify that f(f −1(x)) = x and f −1(f(x)) = x.
Example 3: Find the inverse (if it exists) of
f −1(x) = 0.25x + 1.25
f (x) = 4x − 5 .
III. Derivative of an Inverse Function (Pages 347−348)
Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true.
1. If f is continuous on its domain, then f −1 is continuous on its domain.
2. If f is increasing on its domain, then f −1 is increasing on its domain.
3. If f is decreasing on its domain, then f −1 is decreasing on its domain.
4. If f is differentiable on an interval containing c and f ′(c) ≠ 0, then f −1 is differentiable at f(c).
Let f be a function that is differentiable on an interval I. If f has
an inverse function g, then g is dif f erentiable
at any x f or w h ich f ′ ( g ( x )) ≠ 0 . Moreover,
What you should learn How to find the derivative of an inverse function
g′(x) =
1,f ′(g(x))
f ′(g(x)) ≠ 0 .
This last theorem can be interpreted to mean that graphs
of inverse functions have reciprocal slopes .
10 Chapter Logarithmic, Exponential, and Other Transcendental
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Additional notes
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Section 5.4 Exponential Functions: Differentiation and Integration 103
Section 5.4 Exponential Functions: Differentiation and Integration
Objective: In this lesson you learned about the properties of the natural exponential function and how to find the derivative and antiderivative of the natural exponential function.
I. The Natural Exponential Function (Pages 352−353)
The inverse function of the natural logarithmic function
f (x) = ln x is called the natural exponential
Course Number
Instructor
Date
What you should learn How to develop properties of the natural exponential function
function and is denoted by f −1 (x) = ex . That is,
y = ex if and only if x = ln y .
Example 1: Solve e x−2 − 7 = 59 for x. Round to three decimal places.x ≈ 6.190
Example 2: Solve 4 ln 5x = 28 for x. Round to three decimal places.x ≈ 219.327
Complete each of the following operations with exponential functions.
1. ea eb = ea+b .
2.e
= ea−b .eb
List four properties of the natural exponential function.
1. The domain of f(x) = ex is (−∞, ∞), and the range is (0, ∞).
2. The function f(x) = ex is continuous, increasing, and one-to- one on its entire domain.
3. The graphs of f(x) = ex is concave upward on its entire domain.
10 Chapter Logarithmic, Exponential, and Other Transcendental
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4. lim ex = 0 and lim ex = ∞x→ − ∞ x→ ∞
II. Derivatives of Exponential Functions (Pages 354−355)
Let u be a differentiable function of x. Complete the
following rules of differentiation for the natural exponential
function:
What you should learn How to differentiate natural exponential functions
d ⎡ex ⎤ =
ex .
dx ⎣ ⎦d ⎡eu ⎤ =
eu [du/dx] .
dx ⎣ ⎦
Example 3: Find the derivative of2xex + x2ex
f (x) = x2ex .
III. Integrals of Exponential Functions (Pages
356−357) Let u be a differentiable function of x.What you should learn How to integrate natural exponential functions
∫ex dx
=
∫eu du
=
ex + C
eu + C
Example 4: Find ∫e2 x dx .(1/2) e2x + C
Homework Assignment
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10Section Bases Other Than e and
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Exercises
a
10 Chapter Logarithmic, Exponential, and Other Transcendental
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Section 5.5 Bases Other Than e and Applications
Objective: In this lesson you learned about the properties, derivatives, and antiderivatives of logarithmic and exponential functions that have bases other than e.
I. Bases Other than e (Pages 362−363)
Course Number
Instructor
Date
What you should learnHow to define
If a is a positive real number (a ≠ 1)
and x is any real number, exponential functions that
then the exponential function to the base a is denoted by a x
and is defined by ax = e(ln a)x . If a = 1, then
y = 1x = 1 is a constant function .
In a situation of radioactive decay, half-life is the nu m ber
of y ears required for half of the atoms in a sa m ple of
radioactive mater i al to decay .
have bases other than e
If a is a positive real number (a ≠ 1)
and x is any positive real
number, then the logarithmic function to the base a is denoted
by loga x and is defined by loga x = 1/(ln a ) ln x .
Complete the following properties of logarithmic functions to the base a.
1) loga 1 = 0
2) loga (xy) = l og a x + lo g a y
3. log
xn = n loga x
x4. loga y = l og a x − log a y
State the Properties of Inverse Functions
y = ax if and only if x = loga y
aloga x = x for x > 0
loga ax = x, for all x
10Section Bases Other Than e and
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The logarithmic function to the base 10 is called the com m on
logarith m ic function .
Example 1: (a) Solve log8 x = 1
3
for x.
(b) Solve 5 x = 0.04 for x.(a) x = 2 (b) x = − 2
II. Differentiation and Integration (Pages 364−365)
To differentiate exponential and logarithmic functions to
other bases, you have three options:
1. Use the definitions of ax and loga x and differentiate using the rules for the natural exponential and logarithmic functions.
2. Use logarithmic differentiation, or
3. Use the differentiation rules for bases other than e.
What you should learn How to differentiate and integrate exponential functions that have bases other than e
Let a be a positive real number (a ≠ 1)
and let u be a
differentiable function of x. Complete the following formulas
for the derivatives for bases other than e.d ⎡a x ⎤ =
(ln a) ax .
dx ⎣ ⎦d ⎡au ⎤ =
(ln a) au [du/dx] .
dx ⎣ ⎦d
[log x] = 1 / [ (ln a ) x ] .dx a
d [log u] = 1 / [ (ln a ) u ] [ du / dx ] .
dx a
Occasionally, an integrand involves an exponential function to
a base other than e. When this occurs, there are two options:
(1) convert to base e using the formula ax = e(ln a)x and
then integrate or (2) integrate directly using the
10 Chapter Logarithmic, Exponential, and Other Transcendental
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integration formula ∫a x dx =
(1/ln a) ax + C .
10Section Bases Other Than e and
Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights
Let n be any real number and let u be a differentiable function ofx. The Power Rule for Real Exponents gives.
d [xn ] = nxn−1 .
dxd
[un ] = nun−1 [du/dx] .dx
III. Applications of Exponential Functions (Pages
366−367) Complete the following limit statement:
x x
What you should learn How to use exponential functions to model compound interest and
lim ⎛1 + 1 ⎞ = lim ⎛ x + 1 ⎞ = e . exponential growth
x→∞ ⎜ x ⎟ x→∞ ⎜ x ⎟⎝ ⎠ ⎝ ⎠
Let P be the amount deposited, t the number of years, A the balance after t years, and r the annual interest rate (in decimal form), and n the number of compounding per year. Complete the following compound interest formulas:
Compounded n times per year: A = P(1 + r/n)nt
Compounded continuously: A = Pert
Example 2: Find the amount in an account after 10 years if$6000 is invested at an interest rate of 7%,(a) compounded monthly.(b) compounded continuously.
(a) $12,057.97 (b) $12,082.52
11 Chapter Logarithmic, Exponential, and Other Transcendental
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Exercises
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Section 5.6 Inverse Trigonometric Functions: Differentiation 109
Section 5.6 Inverse Trigonometric Functions: Differentiation
Objective: In this lesson you learned about the properties of inverse trigonometric functions and how to find derivatives of inverse trigonometric functions.
Course Number
Instructor
Date
I. Inverse Trigonometric Functions (Pages 373−375)
None of the six basic trigonometric functions has an
inverse function . This is true because all six
trigonometric functions are p e riodic and th e refore are
not one-to-one . However, their domains can be
redefined in such a way that they will have inverse functions on
th e restricted domains .
For each of the following definitions of inverse trigonometric functions, supply the required restricted domains and ranges.
What you should learn How to develop properties of the six inverse trigonometric functions
y = arcsin x iff sin y = x
D o m a in
− 1 ≤ x ≤ 1
Range
− π /2 ≤ y ≤ π /2
y = arccos x iff cos y = x − 1 ≤ x ≤ 1 0 ≤ y ≤ π
y = arctan x iff tan y = x − ∞ < x < ∞ − π /2 < y < π /2
y = arccot x iff cot y = x − ∞ < x < ∞ 0 < y < π
y = arcsec x iff sec y = x | x | ≥ 1 0 ≤ y ≤ π , y ≠ π /2
y = arccsc x iff csc y = x | x | ≥ 1 − π /2 ≤ y ≤ π /2, y ≠ 0
An alternative notation for the inverse sine function is
sin−1 x .
Example 1: Evaluate the function:− π/2
Example 2: Evaluate the function:
π/3
arcsin (−1) .
arccos 1
.2
Example 3: Evaluate the function: arcos (0.85).0.5548
1 Chapter Logarithmic, Exponential, and Other Transcendental
Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights
State the Inverse Property for the Sine function.
If − 1 ≤ x ≤ 1 and − π/2 ≤ y ≤ π/2, then sin(arcsin x) = x and arcsin(sin y) = y.
State the Inverse Property for the Cosine function.
If − 1 ≤ x ≤ 1 and 0 ≤ y ≤ π, then cos(arccos x) = x and arccos(cos y) = y.
State the Inverse Property for the Tangent function.
If x is a real number and − π/2 < y < π/2, then tan(arctan x) = x and arctan(tan y) = y.
II. Derivatives of Inverse Trigonometric Functions(Pages 376−377)
Let u be a differentiable function of x.
What you should learn How to differentiate an inverse trigonometric function
d [arcsin u] =dx
d [arccos u] =dx
d [arctan u] =dx
d [arc cot u] =dxd [arc secu] =
u′
1 − u2
− u′
1 − u2
u′
1 + u2
− u′
1 + u2
u′
dx
d [arc cscu] =dx
| u |
| u |
u2 − 1
− u′
u2 − 1
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∫
∫
∫
1 Chapter Logarithmic, Exponential, and Other Transcendental
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Section 5.7 Inverse Trigonometric Functions: Integration 111
Section 5.7 Inverse Trigonometric Functions: Integration
Objective: In this lesson you learned how to find antiderivatives of inverse trigonometric functions.
Course Number
Instructor
Date
I. Integrals Involving Inverse Trigonometric Functions
(Pages 382−383)
Let u be a differentiable function of x, and let a > 0.
du
What you should learn How to integrate functions whose antiderivatives involve inverse trigonometricfunctions
∫ a2 −
u2
= arcsin ( u / a ) + C .
du a2 + u2
= ( 1/ a ) arctan ( u / a ) + C .
du
u u2 − a2
Example 1:
= ( 1/ a ) arcs e c (| u | / a ) + C .
6x dx4 + 9x4
½ arctan(3x2/2) + C
II. Completing the Square (Pages 383−384)
Completing the square helps when quadratic
functions are involved in t h e integrand .
Example 2: Complete the square for the polynomial:x2 + 6x + 3 .(x + 3)2 − 6
Example 3: Complete the square for the polynomial: 2x2 + 16x .
2[(x + 4)2 − 16]
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What you should learn How to use the method of completing the square to integrate a function
11 Chapter Logarithmic, Exponential, and Other Transcendental
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∫ ∫
∫
III. Review of Basic Integration Rules (Pages
385−386) Complete the following selected basic
integration rules.
u′ 1
u dx =
u du = ln| u | + C
What you should learn How to review the basic integration rules involving elementary functions
∫du = u + C
∫cot u du = ln|sin u | + C
du a2 + u2
= ( 1/ a ) arctan ( u / a ) + C
∫cos u du = sin u + C
∫sec2 u du = t a n x + C
Homework Assignment
Page(s)
Exercises
sinh x = (ex − e−x) / 2 .
cosh x = (ex + e−x) / 2 .
11Section Hyperbolic
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Section 5.8 Hyperbolic Functions
Objective: In this lesson you learned about the properties of hyperbolic functions and how to find derivatives and antiderivatives of hyperbolic functions.
I. Hyperbolic Functions (Pages 390−392)
Complete the following definitions of the hyperbolic functions.
Course Number
Instructor
Date
What you should learn How to develop properties of hyperbolic functions
tanh x = (sinh x ) / (cosh x ) .
csch x = 1 / (sinh x ), x ≠ 0 .
sech x = 1 / (cosh x ) .
coth x = 1 / (tanh x ), x ≠ 0 .
Complete the following hyperbolic identities.
cosh2 x − sinh2 x = 1 .
tanh2 x + sech2 x = 1 .
coth2 x − csch2 x = 1 .
− 1 + c o sh 2 x 2
= sinh2 x .
1 + co s h 2 x 2 = cosh2 x .
2 sinh x cosh x = sinh 2 x .
cosh2 x + sinh2 x = cosh 2 x .
sinh (x + y)= sinh x c o s h y + cosh x sinh y .
sinh (x − y)= sinh x c o s h y − cosh x sinh y .
cosh (x + y)= cosh x cosh y + sinh x sinh y .
cosh (x − y)= cosh x cosh y − sinh x sinh y .
11 Chapter Logarithmic, Exponential, and Other Transcendental
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II. Differentiation and Integration of Hyperbolic Functions(Pages 392−394)
Let u be a differentiable function of x. Complete each of the following rules of differentiation and integration.
d [sinh u] = (cosh u ) u ′ .
dx
d [cosh u] = (sinh u ) u ′ .
dx
d [tanh u] = (sech2 u) u′ .
dx
d [coth u] = − (csch2 u) u′ .
dx
d [sech u] = − (sech u tanh u ) u ′ .
dx
d [csch u] = − (csch u coth u ) u ′ .
dx
What you should learn How to differentiate and integrate hyperbolic functions
∫cosh u
du
∫sinh u
du
= sinh u + C .
= cosh u + C .
∫sech2 u
du
∫csch2 u
du
= tanh u + C .
= − coth u + C .
∫sech u tanh u
du
= − sech u + C .
∫csch u coth u du = − csch u + C .
11Section Hyperbolic
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III. Inverse Hyperbolic Functions (Pages 394−396)
State the inverse hyperbolic function given by each of the following definitions and give the domain for each.
D o m a in
What you should learn How to develop properties of inverse hyperbolic functions
ln ( x +
ln ( x +
x2 + 1) = sinh−1 x , ( −∞ , ∞ ) .
x2 −1) = cosh−1 x , [ 1, ∞ ) .
1 ln 1 +
x2 1 −
x
= tanh−1 x , ( − 1, 1) .
1 ln
x + 1 = coth−1 x , ( −∞ , − 1) ∪ (1, ∞ ) .
2 x −1
ln 1
+1 − x2
x
= sech−1 x , (0, 1] .
⎛ 1 1 + x2 ⎞ln ⎜+
⎟ = csch−1 x , ( −∞ , 0) ∪ (0, ∞ ) .
⎜ x x ⎟⎝ ⎠
IV. Differentiation and Integration of Inverse Hyperbolic Functions (Pages 396−397)
Let u be a differentiable function of x. Complete each of the following rules of differentiation and integration.
What you should learn How to differentiate and integrate functions involving inverse hyperbolic functions
d [ sinh−1
u] = u′
dx u2 + 1
d [ cosh−1
u ] = u′
dx u2 −1
d [ tanh −1 u ] =dx
u′1 − u2
11 Chapter Logarithmic, Exponential, and Other Transcendental
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d [ coth −1 u ] =dx
u′1 − u2
d [ sech−1
u] = − u ′
dx u 1 − u2
∫
∫
11Section Hyperbolic
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d [ csch−1
u] = − u ′
dx u 1 + u2
du
u2 ±
a2
= ln(u + √u2 ± a2) + C .
du a2 − u2
du
= 1 / (2 a ) ln| ( a + u ) / ( a − u ) | + C .
−(1/a) ln( (a + √a2 ± u2) / |u|) + C .∫ u a2 ± u2 =
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