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 • Simpson’s 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 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.

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 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.

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.x=0 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.x=a is a singular point.

21. Example • Ex. For what values of p>0 is the improper integral convergent? • Sol. x=b 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. Thefunction 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