Integration Using Tables and Summary of Techniques

Transcription

1 Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13

2 Introduction We wrap up integration techniques by discussing the following topics: 1 Integration using tables. 2 How to approximate integrals? 3 Additional tools to find integrals. 4 Summary of Integration Techniques Philippe B. Laval (KSU) Summary Today 2 / 13

3 Integration Using Tables Because integrals are used so often and there are so many different integrals to find, books containing integral formulas exist. However, an integral often needs to be transformed before it fits a particular formula. The techniques used to transform a function so that it will fit a given pattern involve substitution, completing the square and long division. We will illustrate some of these techniques by looking at specific examples.the formulas referred to in these examples come from a list of formulas on my website. Philippe B. Laval (KSU) Summary Today 3 / 13

4 Integration Using Tables Find 9 + x 2 dx Find 9 4x 2 dx Find 4x 2 9dx Find x 2 + 2x + 10dx Philippe B. Laval (KSU) Summary Today 4 / 13

5 Approximation of Definite Integrals Elementary functions: polynomial rational power exponential logarithmic trigonometric inverse trigonometric functions Any functions which can be obtained from these by addition, subtraction, multiplication, division and composition. Question: Can we integrate all the elementary functions that is can we find an antiderivative of all the elementary functions in terms of other elementary functions? Philippe B. Laval (KSU) Summary Today 5 / 13

6 Approximation of Definite Integrals The answer is NO Here are some elementary functions which do not have an antiderivative which can be expressed in terms of elementary functions. cos (e x ) dx e x 2 dx e x x dx sin ( x 2 ) dx dx ln x sin x x What do we do when a problem arises in which we have to evaluate an integral containing one of these functions? dx Philippe B. Laval (KSU) Summary Today 6 / 13

7 Approximation of Definite Integrals Integrals can be approximated using Riemann sums. We already discussed how to do this earlier this semester. There are other techniques which can be used though we will not cover them in this class. Two such techniques can be found in most calculus books. They are: The trapezoidal rule. Simpson s rule. More advanced techniques can also be found in numerical analysis books. Philippe B. Laval (KSU) Summary Today 7 / 13

8 Additional Tools Advanced calculators such as the TI 82, TI 83, TI 86, TI 89, TI 92 as well as many computer software (CAS for Computer Algebra Systems) can evaluate integrals. They usually take one of two approaches. 1 Symbolic integration: This refers to integration by finding an antiderivative first. Tools such as Maple, Mathematica, Mupad, MATLab and the TI 92 can find antiderivatives. 2 Numerical integration: This refers to techniques which approximate integrals using Riemann sums or similar techniques. Only definite integrals can be evaluated. Philippe B. Laval (KSU) Summary Today 8 / 13

9 Summary We have studied the following techniques: 1 Fundamental theorem of Calculus. This changes the problem of finding an integral to the problem of finding antiderivatives. 2 Substitution. 3 Integration by parts. 4 Partial fraction decomposition. 5 Integrals involving trigonometric functions. 6 Trigonometric substitution. 7 Tables of integrals. Philippe B. Laval (KSU) Summary Today 9 / 13

10 Summary Given an integral f (x) dx ( f (x) is called the integrand), try the following steps: 1 Simplify the integrand. 2 Look for an obvious substitution. Try to find some function g (x) in the integrand whose differential g (x) dx also occurs (up to a constant). In this case, try the substitution u = g (x). 3 Classify the integrand according to its form. 1 Trigonometric functions. 2 Rational functions. Try the techniques described in the handout on partial fractions. 3 Integration by parts. If f (x) is a product of a power of x (or a polynomial) and a transcendental function (trigonometric, exponential, logarithmic), try integration by parts. 4 Radicals. If the integral contains a 2 ± x 2 or x 2 a 2, use the appropriate trigonometric substitution. Philippe B. Laval (KSU) Summary Today 10 / 13

11 Summary Try again. If the first three steps have produced no answer, try harder. 1 Try substitution. You may not have tried all the possibilities. 2 Integration by parts. Even if your integral does not have the form described in (3.3), integration by parts may still work. Remember that it works sometimes on f (x) dx with u = f (x) such as sin 1 xdx, tan 1 xdx and ln xdx. 3 Manipulate the integrand. Try to change the integrand using algebraic manipulations, rationalizing. If the integrand involve trigonometric functions, try to rewrite it in terms of other trigonometric functions using identities. 4 Relate the problem to other problems. Think if you have done a similar problem and how you did it. This is why it is important to remember the problems you do and analyze them. 5 Use several methods. Sometimes, several methods may be used. Either you will repeat the same method several times, or you will mix the various methods studied. Philippe B. Laval (KSU) Summary Today 11 / 13

12 Summary Indicate which method you would use, then carry out the integration. Find sin 4 x sec x dx Find x x 3 3x 2 10x dx Find dx x ln x Philippe B. Laval (KSU) Summary Today 12 / 13

13 Exercises See the problems at the end of my notes on integration by substitution. Philippe B. Laval (KSU) Summary Today 13 / 13