ESSENTIAL CALCULUS CH06 Techniques of integration

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ESSENTIAL CALCULUSCH06 Techniques of
integration
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In this Chapter

• 6.1 Integration by Parts
• 6.2 Trigonometric Integrals and Substitutions
• 6.3 Partial Fractions
• 6.4 Integration with Tables and Computer Algebra
  Systems
• 6.5 Approximate Integration
• 6.6 Improper Integrals
• Review

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Chapter 6, 6.1, P307
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Chapter 6, 6.1, P307
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Chapter 6, 6.1, P309
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How to Integrate Powers of sin x and cos x From
Examples 1 4 we see that the following strategy
works
Chapter 6, 6.2, P314
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(i) If the power of cos x is odd, save one cosine
factor and use cos2x1-sin2x to express the
remaining factors in terms of sin x. Then
substitute usin x.
Chapter 6, 6.2, P314
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(ii) If the power of sin x is odd, save one sine
factor and use sin2x1-cos2x to express the
remaining factors in terms of cos x. Then
substitute ucos x.
Chapter 6, 6.2, P314
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(iii) If the powers of both sine and cosine are
even, use the half-angle identities It is
sometimes helpful to use the identity
Chapter 6, 6.2, P314
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How to Integrate Powers of tan x and sec x From
Examples 5 and 6 we have a strategy for two cases
Chapter 6, 6.2, P315
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(i) If the power of sec x is even, save a factor
of sec2x and use sec2x1tan2x to express the
remaining factors in terms of tan x. Then
substitute utan x.
Chapter 6, 6.2, P315
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(ii) If the power of tan x is odd, save a factor
of sec x tan x and use tan2xsec2x-1 to express
the remaining factors in terms of sec x. Then
substitute u sec x.
Chapter 6, 6.2, P315
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Chapter 6, 6.2, P315
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Chapter 6, 6.2, P315
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TABLE OF TRIGONOMETRIC SUBSTITUTIONS
Expression Substitution
Identity
Chapter 6, 6.2, P317
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Chapter 6, 6.3, P326
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Chapter 6, 6.5, P336
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Chapter 6, 6.5, P336
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Chapter 6, 6.5, P336
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If we divide a,b into n subintervals of equal
length ?x(b-a)/n , then we have where X1 is
any point in the ith subinterval xi-1,xi.
Chapter 6, 6.5, P336
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Left endpoint approximation
Chapter 6, 6.5, P336
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Right endpoint approximation
Chapter 6, 6.5, P336
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MIDPOINT RULE
where
and
Chapter 6, 6.5, P336
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Chapter 6, 6.5, P337
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TRAPEZOIDAL RULE
Where ?x(b-a)/n and xiai?x.
Chapter 6, 6.5, P337
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Chapter 6, 6.5, P337
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3. ERROR BOUNDS Suppose f(x)K for axb. If
ET and EM are the errors in the Trapezoidal and
Midpoint Rules, then
and
Chapter 6, 6.5, P339
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Chapter 6, 6.5, P340
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Chapter 6, 6.5, P340
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SIMPSONS RULE
Where n is even and ?x(b-a)/n.
Chapter 6, 6.5, P342
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ERROR BOUND FOR SIMPSONS RULE Suppose that
f(4)(x)K for axb. If Es is the error
involved in using Simpsons Rule, then
Chapter 6, 6.5, P343
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In defining a definite integral we
dealt with a function f defined on a
finite interval a,b. In this section we extend
the concept of a definite integral to the
case where the interval is infinite and also to
the case where f has an infinite discontinuity in
a,b. In either case the integral is called an
improper integral.
Chapter 6, 6.6, P347
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Improper integrals Type1 infinite
intervals Type2 discontinuous integrands
Chapter 6, 6.6, P347
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DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1 (a)
If exists for every number ta,
then provided this limit exists (as a finite
number). (b) If exists for every number
tb, then provided this limit exists (as a
finite number).
Chapter 6, 6.6, P348
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The improper integrals and
are called convergent if the corresponding limit
exists and divergent if the limit does not
exist. (c) If both and are
convergent, then we define In part (c) any
real number can be used (see Exercise 52).
Chapter 6, 6.6, P348
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is convergent if pgt1 and divergent if p1.
Chapter 6, 6.6, P351
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• 3.DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2
• If f is continuous on a,b) and is discontinuous
  at b, then
• if this limit exists (as a finite number).
• (b) If f is continuous on (a,b and is
  discontinuous at a, then
• if this limit exists (as a finite number).

Chapter 6, 6.6, P351
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The improper integral is called
convergent if the corresponding limit exists and
divergent if the limit does not exist (c)If f
has a discontinuity at c, where altcltb, and both
and are convergent, then we
define
Chapter 6, 6.6, P351
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erroneous calculation This is wrong because
the integral is improper and must be calculated
in terms of limits.
Chapter 6, 6.6, P352
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• COMPARISON THEOREM Suppose that f and g are
  continuous functions with f(x)g(x)0 for xa .
• (b) If is divergent, then
  is divergent.
• If is convergent, then
  is convergent.

Chapter 6, 6.6, P353