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 forma. rational functions: partial fractionsb. rational trigonometric functions:c. product of two different kind of functions: integrationby partsd. irrational functions: trigonometric substitution, rationalsubstitution, reciprocal substitution4. Try again. Manipulate the integrand, use severalmethods, relate the problem to known problems
5 Example Integrate Sol I rational substitution works but complicated Sol II manipulate the integrand first
6 ExampleEx. FindSol I. Substitution works but complicatedSol II.
7 Can we integrate all continuous functions? Since continuous functions are integrable, any continuousfunction f has an antiderivative.Unfortunately, we can NOT integrate all continuousfunctions. This means, there exist functions whoseintegration can not be written in terms of essential functions.The typical examples are:
8 Approximate integration In some situation, we can not find An alternativeway is to find its approximate value.By definition, the following approximations are obvious:left endpoint approximationright endpoint approximation
10 Improper integralsThe definite integrals we learned so far are defined on afinite interval [a,b] and the integrand f does not have aninfinite discontinuity.But, to consider the area of the (infinite) region under thecurve from 0 to 1, we need to study the integrabilityof the function on the interval [0,1].Also, when we investigate the area of the (infinite) regionunder the curve from 1 to we need to evaluate
11 Improper integral: type I We now extend the concept of a definite integral to thecase where the interval is infinite and also to the case wherethe integrand f has an infinite discontinuity in the interval. Ineither case, the definite integral is called improper integral.Definition of an improper integral of type I If for anyb>a, f is integrable on [a,b], thenis called the improper integral of type I of f on anddenoted by If the right side limitexists, we say the improper integral converges.
12 Improper integral: type I Similarly we can define the improper integraland its convergence.The improper integral is defined asonly when both and are convergent,the improper integral converges.
13 Example Ex. Determine whether the integral converges or diverges. Sol divergeEx. FindSol.
14 ExampleEx. FindSol.Remark From the definition and above examples, we seethe 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 aproper integral by substitution, then the improper integralis convergent.Ex. EvaluateSol. Let then
17 Improper integral: type II Definition of an improper integral of type II If f iscontinuous on [a,b) and x=b is a vertical asymptote ( b issaid to be a singular point ), thenis 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 theimproper integralIf f has a singular point c inside the interval [a,b], then theOnly when both of the two improper integrals andconverge, the improper integral converge.
19 ExampleEx. FindSol. x=0 is a singular point of lnx.Sol.
20 Example Again, Newton-Leibnitz formula, substitution rule and integration by parts are all true for improper integrals oftype II.Ex. FindSol. 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 ExampleDetermine whether the integral is convergent or divergent
24 Evaluation of improper integrals All integration techniques and Newton-Leibnitz formulahold true for improper integrals.Ex. The function defined by the improper integralis called Gamma function. EvaluateSol.