Presentation on theme: "Techniques of Integration"— Presentation transcript:
1 Techniques of Integration Chapter 7Techniques of Integration
2 7.1 Integration by parts Question: How to integrate , where the integrands are the product of two kinds of functions?Every differentiation rule has a corresponding integration rule:Differentiation Integrationthe Chain Rule the Substitution Rulethe Product Rule the Rule for Integration by Parts
3 The formula for integration by parts (1) Let u = f (x) and v = g(x) are both differentiable, then du= f’(x) dx and dv= g’(x) dx(2)
4 Example 1. FindExample 2. EvaluateExample 3. FindExample 4. Evaluate
5 The formula for definite integration by parts (3) Example 5. CalculateExample 6. Prove the reduction formulawhere is an integer.
6 SummarizeWhen the integrands are the product of two kinds of functions and neither of them is derivative of the other, we use the integration by parts. We can compare:
7 First we recognize u and v, then confirm to be more easily integrated than . For example
8 7.2 Trigonometric integrals Question 1: How to evaluate ？(a) If the power of cosine is odd ( n=2k+1 ).Save one cosine factor toUse to express the remaining factors in terms of sine:Then substitute u=sinx.
9 (b) If the power of sine is odd ( m=2k+1 ). Save one sine factor toUse to express the remaining factors in terms of sine:Then substitute u=cosx.(c) If the powers of both sine and cosine are even, use the half-angle identities
10 It is sometimes helpful to use the identity Example 1. EvaluateExample 2. EvaluateExample 3. Find
11 Question 2: How to evaluate ？ (a) If the power of secant is even ( n=2k).Save a factor ofUse to express the remaining factors in terms of tanx:Then substitute u=tanx.Example 4. Find
12 (b) If the power of tangent is odd ( m=2k+1). Save a factor ofUse to express the remaining factors in terms of secx:Then substitute u=secx.Example 5. Find
13 (c) If n = 0, only tanx occurs. Use and, if necessary, the formula Example 6. Find(d) If n is odd and m is even, we express the integrand completely in term of secx.Power of secx may require integration by parts.Example 7. FindExample 8. Find
14 Question 3: How to evaluate ？ To evaluate the integrals (a)(b)use the corresponding identity:Example 9. Evaluate
15 7.3 Trigonometric substitution How to find the area of a circle or an ellipse?How to integrateIn general we can make a substitution of the form x=g(t) by using the Substitution Rule in reverse(called inverse substitution):Assume that g has an inverse function, that is, g is one-to-one, we obtain
16 Table of trigonometric substitution One kind of inverse substitution is trigonometric substitution.Table of trigonometric substitutionExpression Substitution Identity
17 Example 1. EvaluateExample 2. FindExample 3. EvaluateExample 4. Evaluate
18 Inverse Substitution Formula For Definite Integral Let x=g( t ) and g has an inverse function, we havewhereExample 5. FindExample 6. Find the area enclosed by the ellipseExample 7. Find
19 7.4 Integration of rational functions by partial fractionIn this section we show how to integrate any rational function (a ratio of polynomials) by expressing it as a sum of simpler fraction(called partial fraction).Consider a rational functionwhere P and Q are polynomials. Ifwhere then the degree of P is n and we writedeg(P) = n
20 Step 1. First express f as a sum of simpler fractions. We can integrate rational functions according to 3 steps:Step 1. First express f as a sum of simpler fractions.Provide that the degree of P is less than the degree of Q. Such a rational function is called proper.If f is improper, that is, we can divide Q into P by long division until a remainder R(x) is obtained such that deg(R)<deg(Q). The division statement is(1)where S and R are also polynomials.
21 Step 2. Second factor the denominator Q(x) as far as possible Step 2. Second factor the denominator Q(x) as far as possible. It can be shown that any polynomial Q can be factored as a product of linear factors(of the form ax+b) and irreducible quadratic factors (of the formStep 3. Finally express the proper rational function R(x)/Q(x) as a sum of partial fractions of the formA theorem in algebra guarantees that it is always possible to do it.
22 Case 1. The denominator Q(x) is a product of distinct linear factors. We explain the details for the 4 cases that occur.Case 1. The denominator Q(x) is a product of distinct linear factors.The partial fraction theorem states there exist constantssuch that(2)These constants need to be determined.Example 1. Evaluate
23 Example 2. FindCase 2. Q(x) is a product of linear factors, some of which are repeated.Suppose the first linear factor is repeated r times; that is, occurs in the factorization of Q(x). Then instead of the single termin Equation 2, we would use(3)Example 3. Find
24 Case 3. Q(x) contains irreducible quadratic factors, none of which is repeated. If Q(x) has the factor then, in addition to the partial fraction in Equation 2 and 3, the expression for R(x)/Q(x) will have a term of the form(4)where A and B are constants to be determined. We can integrate (4) by completing the square and using the formula(5)Example 3. Find
25 Case 4. Q(x) contains a repeated irreducible quadratic factors. Example 4. EvaluateCase 4. Q(x) contains a repeated irreducible quadratic factors.If Q(x) has the factor then instead of the single partial fraction(4), the sum(6)occurs in the partial fraction decomposition of R(x)/Q(x). Each of the term in (6) can be integrated by completed the square and making a tangent substitution.Example 5. Evaluate
26 7.5 Rationalizing substitutions By means of appropriate substitutions, some functions can be changed into rational functions. In particular, when an integrand contains an expression of the form , then the substitution u = may be effective.Example 1. EvaluateLetExample 2. FindLet
27 The substitution t = tan(x/2) will convert any rational function of sinx and cosx into an ordinary rational function. This is called Weierstrass substitution.LetThenTherefore
28 (1)Since t = tan(x/2), we have , soThus if we make the substitution t = tan(x/2), then we haveExample 3. Find
29 7.6 Strategy For Integration Integration is more challenging than differentiation.In finding the derivative of a function it is obvious which differentiation formula we should apply. But when integrating a given function, it may not be obvious which techniques we should use.First it is useful to be familiar with the basic integration formulas.Table of integration formulasConstants of integration have been omitted.
31 1. Simplify the integrand if possible. Secondly if you do not immediately see how to attack a given integral, you might try the following four-step strategy.1. Simplify the integrand if possible.2. Look for an obvious substitution.3. Classify the integrand according to its form.4. Try again.(a) Try substitution (b) Try parts.(c) Manipulate the integrand.(d) Relate the problem to previous problems.(e) Use several methods.
32 Example 1.Example 2.Example 3.Example 4.Example 5.
33 Computer Algebra Systems 7.7 Using Tables of Integrals andComputer Algebra Systems
34 7.8 Approximation Integration How to integrateIt is difficult, or even impossible, to find an antiderivative.When the function is determined from a scientific experiment through instrument readings, how to integrate such discrete function?In both cases we need to find approximate values of definite integrals.
35 The left endpoint approximation Using Riemann sumsThe left endpoint approximation(1)The right endpoint approximation(2)(3) Midpoint rule
36 Example 1. Use (a) the Trapezoidal Rule (b) the Midpoint Rule with n=5 to approximate the integral
37 (5) Error bounds Suppose If NoticeThe error in using an approximation is defined to be the amount that needs to be added to the approximation to make it exact..In general, we have(5) Error bounds Suppose Ifand are the errors in the Trapezoidal and Midpoint Rules, then
38 Example 2. (a) Use the Midpoint Rule with n=10 to approximate the integral (b) Give an upper bound for the error involved in this approximation.(6) Simpson’s Rulewhere n is even andExample 3. Use Simpson’s Rule with n=10 to approximation
39 (7) Error bound for Simpson’s Rule Suppose that If is the error involved in using Simpson’s Rule, thenExample 4. (a) Use Simpson’s Rule with n=10 to approximate the integral(b) Estimate the error involved in this approximation.
40 Type 1 Infinite Intervals 7.9 Improper IntegralsIn defining a definite integral we deal with (1) the function f defined on a finite interval [a, b]; (2) f is a bounded function.Question: How to integrate a definite integral when the interval is infinite or when f is unbounded ?Type 1 Infinite IntervalsConsider the infinite region that lies under the curve , above the x-axis and from the line x=1 to infinite, can this area A be infinite?
41 NoticeandSo the area of the infinite region is equal to 1 and we write
42 (1)Definition of An Improper Integral of Type 1 (a) If exists for every number , thenprovide this limit exists(as a finite number).(b) If exists for every number , thenThe improper integrals in (a) and (b) are called convergent if the limit exists and divergent if the limit does not exist.(c) If both and are convergent, then we define
43 Example 1. Determine whether the integral is convergent or divergent.Example 2. EvaluateExample 3. EvaluateExample 4. For what value of p is the integralconvergent?(2) is convergent if p >1 and divergent if
44 Type 2 Discontinuous Integrands (3) Definition of An Improper Integral of Type 2(a) If f is continuous on [a,b) and , thenif this limit exists(as a finite number).(b) If f is continuous on (a,b] and , thenThe improper integrals in (a) and (b) are called convergent if the limit exists and divergent if the limit does not exist.
45 (c) If , where a<c<b, and both and are convergent, then we defineExample 5. FindExample 6. Determine whether converges ordiverges.Example 7. Evaluate if possible.Example 8. Find
46 A Comparison Test For Improper Integrals (4)Comparison Theorem Suppose that f and g are continuous functions with(a) If is convergent, then is convergent.(b) If is divergent, then is divergent.Example 9. Show that (a) is convergent.(b) is divergent.