Integration of irrational functions

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# Integration of irrational functions - PowerPoint PPT Presentation

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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Presentation Transcript
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.
• 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 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

Example
• Integrate
• Sol I rational substitution works but complicated
• Sol II manipulate the integrand first
Example
• Ex. Find
• Sol I. Substitution works but complicated
• Sol II.
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
• 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 integration
• Midpoint rule:
• Trapezoidal rule
• Simpson’s rule
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
• 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 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.

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

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

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

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

convergent?

• Sol. x=b is the singular point. When
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.
Example
• Determine whether the integral is convergent or divergent
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.
Example
• Ex. Find
• Sol.
Homework 19
• Section 7.4: 37, 38, 46, 48
• Section 7.5: 31, 39, 44, 47, 59, 65