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Integration of irrational functions

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Example Determine whether the integral is convergent or divergent Evaluation of improper integrals All integration techniques and Newton-Leibnitz formula hold true ... – PowerPoint PPT presentation

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Title: Integration of irrational functions


1
Integration of irrational functions
  • Rational substitution is the usual way to
    integrate them.
  • Ex. Evaluate
  • Sol. Let then

2
Example
  • Ex. Evaluate
  • Sol.

3
Strategy for integration
  • First of all, remember basic integration
    formulae.
  • Then, try the following four-step strategy
  • 1. Simplify the integrand if possible. For
    example
  • 2. Look for an obvious substitution. For example

4
Strategy for integration
  • 3. Classify the integrand according to its form
  • a. rational functions partial fractions
  • b. rational trigonometric functions
  • c. product of two different kind of
    functions integration
  • by parts
  • d. irrational functions trigonometric
    substitution, rational
  • substitution, reciprocal substitution
  • 4. Try again. Manipulate the integrand, use
    several
  • methods, relate the problem to known
    problems

5
Example
  • Integrate
  • Sol I rational substitution works
    but complicated
  • Sol II manipulate the integrand first

6
Example
  • Ex. Find
  • Sol I. Substitution works but
    complicated
  • Sol II.

7
Can we integrate all continuous functions?
  • Since continuous functions are integrable, any
    continuous
  • function f has an antiderivative.
  • Unfortunately, we can NOT integrate all
    continuous
  • functions. This means, there exist functions
    whose
  • integration can not be written in terms of
    essential functions.
  • The typical examples are

8
Approximate integration
  • In some situation, we can not find
    An alternative
  • way is to find its approximate value.
  • By definition, the following approximations are
    obvious
  • left endpoint approximation
  • right endpoint approximation

9
Approximate integration
  • Midpoint rule
  • Trapezoidal rule
  • Simpsons rule

10
Improper integrals
  • The definite integrals we learned so far are
    defined on a
  • finite interval a,b and the integrand f does
    not have an
  • infinite discontinuity.
  • But, to consider the area of the (infinite)
    region under the
  • curve from 0 to 1, we need to
    study the integrability
  • of the function on the interval
    0,1.
  • Also, when we investigate the area of the
    (infinite) region
  • under the curve from 1 to
    we need to evaluate

11
Improper integral type I
  • We now extend the concept of a definite integral
    to the
  • case where the interval is infinite and also to
    the case where
  • the integrand f has an infinite discontinuity in
    the interval. In
  • either case, the definite integral is called
    improper integral.
  • Definition of an improper integral of type I If
    for any
  • bgta, f is integrable on a,b, then
  • is called the improper integral of type I of f on
    and
  • denoted by
    If the right side limit
  • exists, we say the improper integral converges.

12
Improper integral type I
  • Similarly we can define the improper integral

  • and its convergence.
  • The improper integral is
    defined as
  • only when both and
    are convergent,
  • the improper integral
    converges.

13
Example
  • Ex. Determine whether the integral
    converges or diverges.
  • Sol.
    diverge
  • Ex. Find
  • Sol.

14
Example
  • Ex. Find
  • Sol.
  • Remark From the definition and above examples, we
    see
  • the New-Leibnitz formula for improper integrals
    is also true

15
Example
  • Ex. Evaluate
  • Sol.
  • Ex. For what values of p is the integral
    convergent?
  • Sol. When

16
Example
  • All the integration techniques, such as
    substitution rule,
  • integration by parts, are applicable to improper
    integrals.
  • Especially, if an improper integral can be
    converted into a
  • proper integral by substitution, then the
    improper integral
  • is convergent.
  • Ex. Evaluate
  • Sol. Let then

17
Improper integral type II
  • Definition of an improper integral of type II If
    f is
  • continuous on a,b) and xb is a vertical
    asymptote ( b is
  • said to be a singular point ), then
  • is called the improper integral of type II. If
    the limit exists,
  • we say the improper integral converges.

18
Improper integral type II
  • Similarly, if f has a singular point at a, we can
    define the
  • improper integral
  • If f has a singular point c inside the interval
    a,b, then the
  • improper integral
  • Only when both of the two improper integrals
    and
  • converge, the improper integral
    converge.

19
Example
  • Ex. Find
  • Sol. x0 is a singular point of lnx.
  • Ex. Find
  • Sol.

20
Example
  • Again, Newton-Leibnitz formula, substitution rule
    and
  • integration by parts are all true for
    improper integrals of
  • type II.
  • Ex. Find
  • Sol. xa is a singular point.

21
Example
  • Ex. For what values of pgt0 is the improper
    integral
  • convergent?
  • Sol. xb is the singular point. When

22
Comparison test
  • Comparison principle Suppose that f and g are
  • continuous functions with
    for then
  • (a)If converges, then
    converges.
  • (b)If the latter diverges, then the former
    diverges.
  • Ex. Determine whether the integral
    converges.
  • Sol.

23
Example
  • Determine whether the integral is convergent or
    divergent

24
Evaluation of improper integrals
  • All integration techniques and Newton-Leibnitz
    formula
  • hold true for improper integrals.
  • Ex. The function defined by the improper
    integral
  • is called Gamma function. Evaluate
  • Sol.

25
Example
  • Ex. Find
  • Sol.

26
Homework 19
  • Section 7.4 37, 38, 46, 48
  • Section 7.5 31, 39, 44, 47, 59, 65
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