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Integration of irrational functions. Rational substitution is the usual way to integrate them. Ex. Evaluate Sol. Let then . Example. Ex. Evaluate Sol. Strategy for integration. First of all, remember basic integration formulae.

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integration of irrational functions
Integration of irrational functions
  • Rational substitution is the usual way to integrate them.
  • Ex. Evaluate
  • Sol. Let then
example
Example
  • Ex. Evaluate
  • Sol.
strategy for integration
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:
strategy for integration1
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

example1
Example
  • Integrate
  • Sol I rational substitution works but complicated
  • Sol II manipulate the integrand first
example2
Example
  • Ex. Find
  • Sol I. Substitution works but complicated
  • Sol II.
can we integrate all continuous functions
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:
approximate integration
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

approximate integration1
Approximate integration
  • Midpoint rule:
  • Trapezoidal rule
  • Simpson’s rule
improper integrals
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

improper integral type i
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

b>a, 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.

improper integral type i1
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.

example3
Example
  • Ex. Determine whether the integral converges or diverges.
  • Sol. diverge
  • Ex. Find
  • Sol.
example4
Example
  • Ex. Find
  • Sol.
  • Remark From the definition and above examples, we see

the New-Leibnitz formula for improper integrals is also true:

example5
Example
  • Ex. Evaluate
  • Sol.
  • Ex. For what values of p is the integral convergent?
  • Sol. When
example6
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
improper integral type ii
Improper integral: type II
  • Definition of an improper integral of type II If f is

continuous on [a,b) and x=b 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.

improper integral type ii1
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.

example7
Example
  • Ex. Find
  • Sol.x=0 is a singular point of lnx.
  • Ex. Find
  • Sol.
example8
Example
  • Again, Newton-Leibnitz formula, substitution rule and

integration by parts are all true for improper integrals of

type II.

  • Ex. Find
  • Sol.x=a is a singular point.
example9
Example
  • Ex. For what values of p>0 is the improper integral

convergent?

  • Sol. x=b is the singular point. When
comparison test
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.
example10
Example
  • Determine whether the integral is convergent or divergent
evaluation of improper integrals
Evaluation of improper integrals
  • All integration techniques and Newton-Leibnitz formula

hold true for improper integrals.

Ex. Thefunction defined by the improper integral

is called Gamma function. Evaluate

  • Sol.
example11
Example
  • Ex. Find
  • Sol.
homework 19
Homework 19
  • Section 7.4: 37, 38, 46, 48
  • Section 7.5: 31, 39, 44, 47, 59, 65