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