周次: 5 日期: March 23, 2022 节次: 1

In this lecture, we define the inverse trigonometric functions, and develop some basic properties. The properties we are focusing on are the differentiation and integration properties. A brief introduction on hyperbolic functions, in particular the hyperbolic cosine and hyperbolic sine functions are given.

As the second half, we discuss some integration techniques. Those techniques could help us to find more anti-derivatives and as a result, evaluate more definite integrals.

§ 7.7 Inverse Trigonometric Functions

The six basic trigonometric functions are not one-to-one. However, we can restrict their domain to intervals on which they are one-to-one. The sine function increases from 1 at x=π/2 to 1 when x=π/2. By restricting its domain to the interval [π/2,π/2] we make it one-to-one, so it has an inverse sin1(x).

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The cosine function is one-to-one on [0,π].

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The tangent function is one-to-one on (π/2,π/2).

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The cotangent function is one-to-one on (0,π).

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The secant function is one-to-one on [0,π/2)(π/2,π].

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The cosecant function is one-to-one on [π/2,0)(0,π/2].

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Those restricted trigonometric functions have inverses, denoted by (arc). So we have arcsin(x), arccos(x), arctan(x), arccot(x), arcsec(x) and arccsc(x).

The Arcsine and Arccosine Functions

Definition. Arcsine and Arccosine Functions

  • y=sin1(x)=arcsin(x) is the angle in [π/2,π/2] whose sine is x.
  • y=cos1(x)=arccos(x) is the angle in [0,π] whose cosine is x.

Sine (blue) and arcsine (yellow) functions

Sine (blue) and arcsine (yellow) functions

The graph of y=arcsin(x) is symmetric about the origin, therefore an odd function.

arcsin(x)=arcsin(x).

Cosine (blue) and arccosine (yellow) functions

Cosine (blue) and arccosine (yellow) functions

The graph of y=arccos(x) does not have symmetry about the origin or about the y-axis.

Example 1. Common values of arcsine function

  • arcsin(12)=π6 ; arcsin(32)=π3;arcsin(22)=π4.
  • arcsin(0)=0; arcsin(1)=π2; arcsin(1)=π2.

Example 2. Common values of arccosine function

  • arccos(12)=π3; arccos(32)=π6; arccos(22)=π4.
  • arccos(0)=π2; arccos(1)=0; arccos(1)=π.

Identities Involving Arcsine and Arccosine

Basic Identities

Since cos(πx)=cos(x), we have

arccos(x)=πarccos(x),orarccos(x)+arccos(x)=π.

Since cos(π2x)=sin(x), we have

arcsin(x)=π2arccos(x),orarcsin(x)+arccos(x)=π2.

Inverses of tan(x), cot(x), sec(x) ,and csc(x)

Definition. Arctangent and Arccotangent Function

  • y=arctan(x) is the angle in (π/2,π/2) for which tan(y)=x
  • y=arccot(x) is the angle in (0,π) for which cot(y)=x.

The graph of arctan(x)

The graph of arctan(x)

The graph of arctan(x) is symmetric with respect to the origin. Namely, And arccot(x) has no such symmetries.

tan(x)=tan(x).

The graph of arccot(x)

The graph of arccot(x)

Example 3. Common Values of Arctangent Function

  • arctan(13)=π6; arctan(±3)=±π3.

Example 4. Find cos(α), tan(α), sec(α), if

α=arcsin(23)

Example 5. Find sec(arctanx3)

The Derivative of Arcsine Function

Find ddxarcsin(x) by using the formula

ddxf1(x)=1f(f1(x))

The Chain Rule shows

ddxarcsin(g(x))=

Example 7. Applying the Derivative Formula

ddxarcsin(x2)=

The Derivative of Arctangent Function

Find ddxarctan(x)=

Derivatives of the Other Two

Inverse Function-Inverse Cofunction Identities

arccos(x)=π2arcsin(x), arccot(x)=π2arctan(x)

Thus

ddxarccos(x)=11x2, ddxarccot(x)=11+x2

Example 10. A Tangent Line to the Arccotangent Curve

Find an equation for the line tangent to the graph of y=arccot(x) at x=1.

Derivatives Table

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Integration Formulas

The Integration Formulas

11x2dx=arcsin(x)+C

11+x2dx=arctan(x)+C

For integrations like 1a2x2dx and 1a2+x2dx, we can use substitution of variables

1a2x2dx=1a1(x/a)2dx=d(x/a)1(x/a)2=arcsin(x/a)+C

Similarly,

1a2+x2dx=1aarctan(x/a)+C.

Example 11. Using the Integral Formulas

  1. 2/23/2dx1x2=π2
  2. 01dx1+x2=π4
  3. dx9x2
  4. dx34x2

Example 13. Completing the Square

  • Evaluate

dx4xx2.

  • Evaluate

dx4x2+4x+2.

Example 15. Using Substitution

Evaluate

dxe2x6.

§ 7.8 A Brief Introduction on Hyperbolic Functions

Let’s recall the odd and even functions. We have stated that for every function f defined on an interval centered at the origin can be written in a unique way as the sum of one even function and one odd function. The decomposition is

$$ f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}{\text{even part}} + \underbrace{\frac{f(x)-f(-x)}{2}}{\text{odd part}} $$

No exception for exponential function ex , it can be decomposed. And we assigned two new functions to the decomposition.

cosh(x):=even part of ex=ex+ex2

sinh(x):=odd part of ex=exex2

cosh(x) is pronounced as “kosh x”, rhyming with “gosh x”, and sinh(x) is pronounced as “cinch x”, rhyming with “pinch x”.

Graphs of Hyperbolic Functions

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Identities for Hyperbolic Functions

cosh2(x)sinh2(x)=1

sinh(2x)=2sinh(x)cosh(x)

cosh(2x)=cosh2(x)+sinh2(x)

Other Hyperbolic Functions

Analogous to trigonometric functions, the tanh(x), coth(x), sech(x) and csch(x) can all be defined, following the same pattern.

tanh(x):=sinh(x)cosh(x),coth(x):=cosh(x)sinh(x),etc.

The Derivative and Anti-derivative of Hyperbolic Functions

ddxsinh(x)=cosh(x),ddxcosh(x)=sinh(x)

Therefore the anti-derivatives read

sinh(x)=cosh(x)+C,cosh(x)=sinh(x)+C.

Chapter 8. Techniques of Integration

In this chapter we study a number of important techniques for finding indefinite integrals of more complicated functions

§ 8.1 Basic Integration Formulas

Try to match integrals that confront us against one of the standard types. This usually involves a certain amount of algebraic manipulation as well as use of the substitution rules.

Recall that the substitution rule is

f(g(x))g(x)dx=f(u)du,

where u=g(x) is a differentiable function whose range is an interval I and f is continuous on I. Success in integration often hinges on the ability to spot what part of the integrand should be called u in order that one will also have du, so that a known formula can be applied. This means that the first requirement for skill in integration is a thorough mastery of the formulas for differentiation.

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**Note** Students should have a good mastery of formulas 12, 13, 18, 19. For 20 and 22, we provide an alternative (equivalent) anti-derivative

duu2±a2=ln|u+u2±a2|+C

For 20, we have an alternative way to show the anti-derivative, by applying the integration table.

duuu2a2=duu21(au)2=1ad(au)1(au)2=1aarcsin(au)+C

Example 1. Making a Simplifying Substitution

Evaluate

2x9x29x+1dx

Example 2. Completing the Square

Evaluate

dx8xx2dx

Example 3. Expanding a Power and Using a Trigonometric Identity

Evaluate

(secx+tanx)2dx.

Example 4. Eliminating a Square Root

Evaluate

0π/41+cos4x,dx.

Example 5. Reducing an Improper Fraction

Evaluate

3x27x3x+2,dx.

Example 6. Separating a Fraction

Evaluate

3x+21x2,dx.

Example 7. Integral of y=sec(x) — Multiplying by a Form of 1

Evaluate

secx,,dx

Similarly, one can find

cscx,,dx

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