Indefinite Integrals

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Unit 3

• Indefinite Integrals

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3.1 Definition of Indefinite Integrals
For a given function f(x), if F(x) is such that F
(x) f(x), then F(x) is called a primitive or
an antiderivative of f(x).
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3.1 Definition of Indefinite Integrals
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3.1 Definition of Indefinite Integrals
Let F(x) be a primitive of f(x). If G(x) is any
one of the primitives of f(x), then G(x) and F(x)
only differ by a constant.
Every primitive of f(x) can be expressed in the
form F(x) C, where F(x) is a primitive of f(x)
and C is an arbitrary constant.
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3.1 Definition of Indefinite Integrals
Not always true
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3.2 Integration Formulas for Standard Forms
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3.2 Integration Formulas for Standard Forms
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3.2 Integration Formulas for Standard Forms
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3.3 Integration by change of Variable
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3.3 Integration by change of Variable
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3.3 Integration by change of Variable
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3.3 Integration by change of Variable
The form of the answer of an indefinite integral
may not be unique. Both answers are regarded the
same if they only differ by a constant. For
instance, when integrating a function one gives
the answer while others obtain
another answer lnxC. In fact, these two
answers can be regarded the same because they
only differ by a constant ln2.
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P.70 Ex.3A
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3.4 Integration of Trigonometric Functions
(A) Integration of sinmx cosnx, cosmx cosnx,
sinmx sinnx
Use product to sum formulas as the first step
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3.4 Integration of Trigonometric Functions
(B) Integration of sinnx and cosnx
Use sin2? cos2? 1 as the first step if n is
odd.
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3.4 Integration of Trigonometric Functions
(C) Integration of sinmx cosnx
Use sin? d? -d cos?, cos? d? d sin? as the
first step if either m or n is odd or both are
odd.
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3.4 Integration of Trigonometric Functions
(D) Integration of tannx, cotnx, secnx and cscnx
Use tan2? sec2? - 1 as the first step
Use cot2? csc2? - 1 as the first step
Use sec2? tan2? 1 as the first step if n is
even.
Use integration by parts as the first step if n
is odd.
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3.4 Integration of Trigonometric Functions
(E) Integration of tanmx secnx and cotmx cscnx
No generalized first step.
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P.84 Ex. 3B
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3.5 Integration by Parts
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3.5 Integration by Parts
Technique of applying integration by parts
regard the integrand a product of two parts
integrate g(x), getting G(x)
not readily to be integrated
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3.5 Integration by Parts
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P.88 Ex.3C
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3.6 Integration by Substitutions
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3.6 Integration by Substitutions
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P.92 Ex.3D
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3.7 Integration of Rational Functions
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3.7 Integration of Rational Functions
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3.7 Integration of Rational Functions
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P.96 Ex.3E
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3.8 Integration of Rational Functions of sin? and
cos?
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3.8 Integration of Rational Functions of sin? and
cos?
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3.8 Integration of Rational Functions of sin? and
cos?
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3.8 Integration of Rational Functions of sin? and
cos?
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3.8 Integration of Rational Functions of sin? and
cos?
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P.98 Ex.3F
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3.9 Reduction Formulas
Consider the integral
where n is a positive integer. We can easily
evaluate In if n 1, 2, 3 or 4 but for larger
values of n evaluation would be more difficult.
In general, it is difficult to evaluate an
integral which involves terms with higher degrees
and the integral cannot be evaluated by using
standard formulae. However in some cases, where m
and n are positive integers, it is possible to
establish a relation In and Im for n gt m.
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3.9 Reduction Formulas
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3.9 Reduction Formulas
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P.105 Ex.3G