Calculus. Integration (III)

Transcription

1 Calculus Integration (III)

2 Outline 1 Other Techniques of Integration Partial Fractions Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution 2 Using Tables of Integrals Integration Using a Computer Algebra System

3 Outline 1 Other Techniques of Integration Partial Fractions Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution 2 Using Tables of Integrals Integration Using a Computer Algebra System

4 An Overview There are many techniques of integration other than substitution and integration by parts. In this section, we introduce several additional techniques.

5 Partial Fractions We begin with a simple observation. Note that 3 x (x 5) 2(x + 2) = = x 5 (x + 2)(x 5) x 19 x 2 3x 10. So x 19 x 2 3x 10 dx = ( 3 x ) dx x 5 = 3 ln x ln x 5 + c. The second integrand, 3 x x 5 is called a partial fractions decomposition of the first integrand.

6 Partial Fractions: Distinct Linear Factors (I) The above example is a special case of partial fractions. More generally, if the three factors a 1 x + b 1, a 2 x + b 2 and a 3 x + b 3 are all distinct (i.e., no one is a constant multiple of another), then we can write a 1 x + b 1 (a 2 x + b 2 )(a 3 x + b 3 ) = A B +, a 2 x + b 2 a 3 x + b 3 for some choice of constants A and B to be determined. Notice that if we wanted to integrate this expression, the partial fractions on the right-hand side are very easy to integrate, just as they were in the introductory example above.

7 Partial Fractions: Distinct Linear Factors (II) We can do the same as we did above whenever a rational expression has a denominator that factors into n distinct linear factors, as follows. If the degree of P(x) < n and the factors (a i x + b i ), for i = 1, 2,..., n are all distinct, then we can write P(x) n (a 1 x + b 1 )(a 2 x + b 2 ) (a n x + b n ) = = c 1 a 1 x + b 1 + for some constants c 1, c 2,..., c n. c 2 a 2 x + b i=1 c i a i x + b i c n a n x + b n,

8 Partial Fractions: Distinct Linear Factors (III) Example (7.1) 1 Evaluate x 2 + x 2 dx.

9 Partial Fractions: Three Distinct Linear Factors Example (7.2) 3x 2 7x 2 Evaluate x 3 dx. x

10 Partial Fractions Where Long Division Is Required Example (7.3) Find the indefinite integral of f (x) = 2x3 4x 2 15x + 5 x 2 using a 2x 8 partial fractions decomposition.

11 Partial Fractions with a Repeated Linear Factor (I) We now consider the case where the denominator of a rational expression contains repeated linear factors. If the degree of P(x) < n, then we can write P(x) (ax + b) n = n k=1 c k (ax + b) k = c 1 ax + b + c 2 (ax + b) c n (ax + b) n, for constants c 1, c 2,..., c n to be determined.

12 Partial Fractions with a Repeated Linear Factor (II) Example (7.4) Use a partial fractions decomposition to find an antiderivative of f (x) = 5x2 + 20x + 6 x 3 + 2x 2 + x.

13 Partial Fractions: Irreducible Quadratic Factor We can extend the notion of partial fractions decomposition to rational expressions with denominators containing irreducible quadratic factors (i.e., quadratic factors that have no real factorization). If the degree of P(x) < 2n (the degree of the denominator) and each of the factors in the denominator are distinct, then we can write P(x) (a 1 x 2 + b 1 x + c 1 )(a 2 x 2 + b 2 x + c 2 ) (a n x 2 + b n x + c n ) = A 1 x + B 1 (a 1 x 2 + b 1 x + c 1 ) + A 2 x + B 2 (a 2 x 2 + b 2 x + c 2 ) + + A n x + B n (a n x 2 + b n x + c n ).

14 Partial Fractions with a Quadratic Factor (I) Example (7.5) Use a partial fractions decomposition to find an antiderivative of f (x) = 2x2 5x + 2 x 3. + x

15 Partial Fractions with a Quadratic Factor (II) Often, partial fractions decompositions involving irreducible quadratic terms lead to expressions that require further massaging (such as completing the square) before we can find an antiderivative. We illustrate this in example 7.6.

16 Partial Fractions with a Quadratic Factor (III) Example (7.6) Use a partial fractions decomposition to find an antiderivative for f (x) = 5x 2 + 6x + 2 (x + 2)(x 2 + 2x + 5).

17 Outline 1 Other Techniques of Integration Partial Fractions Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution 2 Using Tables of Integrals Integration Using a Computer Algebra System

18 Integrals Involving Powers of Trigonometric Functions Evaluating an integral whose integrand contains powers of one or more trigonometric functions often involves making a clever substitution. These integrals are sufficiently common that we present them here as a group.

19 sin m x cos n x dx Our first aim is to evaluate integrals of the form sin m x cos n x dx, where m and n are positive integers.

20 sin m x cos n x dx Case 1: m or n Is an Odd Positive Integer If m is odd, then we rewrite the integral as follows. sin m x cos n x dx = sin m 1 x cos n x sin xdx = (1 cos 2 ) m 1 2 cos n x d(cos x)

21 sin m x cos n x dx Case 1: m or n Is an Odd Positive Integer Likewise, if n is odd, then we rewrite the integral as follows. sin m x cos n x dx = sin m x cos n 1 x cos xdx = sin m x(1 sin 2 x) n 1 2 d(sin x)

22 sin m x cos n x dx Case 2: m and n Are Both Even Positive Integers In this case, we can use the half-angle formulas for sine and cosine, sin 2 x = cos 2 x = 1 cos 2x cos 2x 2 to reduce the powers in the integrand. We illustrate this case in example 7.9.

23 An Integrand with an Odd Power of Sine Example (7.7) Evaluate cos 4 x sin 3 x dx.

24 An Integrand with an Odd Power of Cosine Example (7.8) sin Evaluate x cos 5 x dx.

25 An Integrand with an Even Power of Sine Example (7.9) Evaluate sin 2 x dx.

26 tan m x sec n x dx (I) Our next aim is to devise a strategy for evaluating integrals of the form sin m x cos n x dx, where m and n are positive integers.

27 tan m x sec n x dx (II) Case 1: m Is an Odd Positive Integer 1 Isolate one factor of sec x tan x. (We ll need this for du.) 2 Replace any factors of tan 2 x with sec 2 x 1 and make the substitution u = sec x. Case 2: n Is an Even Positive Integer 1 Isolate one factor of sec 2 x. (We ll need this for du.) 2 Replace any remaining factors of sec 2 x with 1 + tan 2 x and make the substitution u = tan x.

28 An Integrand with an Odd Power of Tangent Example (7.10) Evaluate tan 3 x sec 3 x dx.

29 An Integrand with an Even Power of Secant Example (7.11) Evaluate tan 2 x sec 4 x dx.

30 Outline 1 Other Techniques of Integration Partial Fractions Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution 2 Using Tables of Integrals Integration Using a Computer Algebra System

31 Trigonometric Substitution (I) If an integral contains a term of the form a 2 x 2, a 2 + x 2 or x 2 a 2, for some a > 0, we can often evaluate the integral by making a substitution involving a trig function (hence, the name trigonometric substitution).

32 Trigonometric Substitution (II) In particular, suppose that an integrand contains a term of the form a 2 x 2, for some a > 0. If we let x = a sin θ, where π 2 θ π 2, then we can eliminate the square root, as follows. Notice that we now have a 2 x 2 = a 2 (a sin θ) 2 = a 2 a 2 sin 2 θ = a 1 sin 2 θ = a cos 2 θ = a cos θ, since for π 2 θ π, cos θ 0. 2

33 Trigonometric Substitution (III) Similarly, if an integrand contains a term of the form a 2 + x 2, for some a > 0, we let x = a tan θ, where π 2 θ π 2. Notice that in this case, we can eliminate the square root, as follows: a 2 + x 2 = a 2 + (a tan θ) 2 = a 2 + a 2 tan 2 θ = a 1 + tan 2 θ = a sec 2 θ = a sec θ, since for π 2 < θ < π, sec θ > 0. 2

34 Trigonometric Substitution (IV) Finally, if an integrand contains a term of the form x 2 a 2, for some a > 0, we let and use the identity x = a sec θ, where θ to eliminate the square root. sec 2 θ 1 = tan 2 θ [ 0, π ) ( π ] 2 2, π,

35 An Integral Involving a 2 x 2 Example (7.12) 1 Evaluate x 2 4 x dx. 2 Figure: [4.26]

36 Outline 1 Other Techniques of Integration Partial Fractions Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution 2 Using Tables of Integrals Integration Using a Computer Algebra System

37 Overviev Integration tables and computer algebra systems are extremely powerful tools for the professional user of mathematics. To use a table, we often must first rewrite the integral in the form of one of the integrals in the table. This may require you to perform some algebraic manipulation or to make a substitution. While a CAS will report an antiderivative, it will occasionally report it in an inconvenient form. More significantly, a CAS will from time to time report an answer that is (at least technically) incorrect.

38 Important Reference 1 A small table of indefinite integrals is included at the back of the book. 2 A larger table can be found in the CRC Standard Mathematical Tables. 3 An amazingly extensive table can be found in the book Table of Integrals, Series and Products, compiled by Gradshteyn and Ryzhik. If you can t find an integral there, you re not likely to find it anywhere.

39 Using an Integral Table Example (8.1) 3 + 4x 2 Use a table to evaluate dx. x Hint: a 2 + x 2 du = a u 2 + u 2 a + a a ln 2 + x 2 u + c

40 Reduction Formulas A number of the formulas in the table are called reduction formulas. These are of the form f (u) du = g(u) + h(u) du, where the second integral is simpler than the first. These are often applied repeatedly, as in example 8.2.

41 Using a Reduction Formula (I) Example (8.2) Use a reduction formula to evaluate sin 6 x dx. Hint: Use the reduction formula sin n u du = 1 n sinn 1 u cos u + n 1 n sin n 2 u du

42 Using a Reduction Formula (II) There are many different ways to find an antiderivative. Antiderivatives found through different means may look quite different, even though they are equivalent. For instance, notice that if an antiderivative has the form sin 2 x + c, then an equivalent antiderivative is cos 2 x + c, since we can write sin 2 x + c = 1 cos 2 x + c = cos 2 x + (1 + c). Finally, since c is an arbitrary constant, so is 1 + c.

43 Using a Reduction Formula (III) In example 8.2, observe that the first three terms all have factors of sin x cos x, which equals 1 sin 2x. 2 Using this and other identities, we can show that the solution in example 8.2 is equivalent to the following solution obtained from a popular CAS: sin 6 x dx = 5 16 x sin 2x sin 4x sin 6x + c

44 Making a Substitution Before Using a Reduction Formula Example (8.3) Evaluate x 3 sin 2x dx. Hint: The reduction formula u n sin u du = u n cos u + n u n 1 cos u du

45 Making a Substitution Before Using an Integral Table As we ll see in example 8.4, some integrals require some insight before using an integral table Example (8.4) sin 2x Evaluate dx. 4 cos x 1 Hint: u = cos x and the formula u du = 2 a + bu 3b 2 (bu 2a) a + bu + c

46 Outline 1 Other Techniques of Integration Partial Fractions Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution 2 Using Tables of Integrals Integration Using a Computer Algebra System

47 Integration Using a Computer Algebra System Computer algebra systems are some of the most powerful new tools to arrive on the mathematical scene in the last 20 years. Powerful software systems like Mathematica, Maple and Math Cad, can run on nearly any personal computer. Here, we point out some mistakes that we may encounter using a CAS.

48 A Shortcoming of Some Computer Algebra Systems Example (8.5) Use a computer algebra system to evaluate Many CASs evaluate instead ln x 1 dx = ln x x 1 x dx.

49 An Incorrect Antiderivative (I) Sometimes the antiderivative reported by a CAS is not valid, written, for any real values of x, as in example 8.6. In some cases, CASs give an antiderivative that is correct for the more advanced case of a function of a complex variable.

50 An Incorrect Antiderivative (II) Example (8.6) Use a computer algebra system to evaluate One CAS reports the incorrect antiderivative cos x dx = ln(sin x 2) sin x 2 The correct antiderivative is ln(2 sin x) + c cos x sin x 2 dx.

51 A Problem Where the CAS Misinterprets What You Enter (I) Probably the most common errors we will run into are actually our own. If we give our CAS a problem in the wrong form, it will solve a different problem than we intended. One simple, but common, mistake is shown in example 8.7.

52 A Problem Where the CAS Misinterprets What You Enter (II) Example (8.7) Use a computer algebra system to evaluate 4x8x dx. After entering the integrand as 4x8x, one CAS returned the odd answer 4x8x dx = 4x8xx The CAS interpreted the integrad as 4 times a variable named x8x, which is unrelated to variable of integration, x.

53 An Inconvenient Form of an Antiderivative The form of the antiderivative reported by a CAS will not always be the most convenient. Example (8.8) Use a computer algebra system to evaluate x ( x ) 5 dx. Several CASs evaluate x ( x 2 + 3) 5 dx = 1 12 x x x8 + 45x x x2, while others return the much simpler expression x ( x 2 + 3) 5 dx = (x2 + 3) 6 12

54 Some Good Integrals for Using a Computer Algebra System (I) Typically, a CAS will perform even lengthy integrations with ease. Example (8.9) Use a computer algebra system to evaluate x 10 sin 2x dx. x 3 sin 2x dx and

55 Some Good Integrals for Using a Computer Algebra System (II) Using a CAS, we can get it in one step x 3 sin 2x dx = 1 2 x3 cos 2x x2 sin 2x 3 8 sin 2x + 3 x cos 2x + c 4 With the same effort, we can obtain x 10 sin 2x dx = 1 2 x10 cos 2x x9 sin 2x x8 cos 2x 45x 7 sin 2x x6 cos 2x x5 sin 2x cos 2x x sin 2x + c 4 2 x3 sin 2x x 2 cos 2x 4

56 A Very Hard Integral (I) A CAS can perform repetitive calculations (numerical or symbolic) that we could never dream of doing by hand. It is difficult to find a function that has an elementary antiderivative that our CAS cannot find. Consider the following example of a hard integral.

57 A Very Hard Integral (II) Example (8.10) Evaluate x 7 e x sin x dx. One CAS reports the antiderivative x 7 e x sin x dx = + ( x7 ( x x6 2 21x x 3 315x x x x4 2 ) e x cos x ) 105x x 315 e x sin x.