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 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. 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. The function 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