Techniques of Integration 1

8 Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. For example, faced with Z x10 dx we realize immediately that the derivative of x11 will supply an x10 : (x11 )′ = 11x10 . We don’t want the “11”, but constants are easy to alter, because differentiation “ignores” them in certain circumstances, so d 1 11 1 x = 11x10 = x10 . dx 11 11 From our knowledge of derivatives, we can immediately write down a number of antiderivatives. Here is a list of those most often used: Z xn+1 + C, if n 6= −1 xn dx = n+1 Z x−1 dx = ln |x| + C Z ex dx = ex + C Z sin x dx = − cos x + C 163 164 Chapter 8 Techniques of Integration Z Z Z cos x dx = sin x + C sec2 x dx = tan x + C sec x tan x dx = sec x + C 1 dx = arctan x + C 1 + x2 Z 1 √ dx = arcsin x + C 1 − x2 Z 8.1 Substitution Needless to say, most problems we encounter will not be so simple. Here’s a slightly more complicated example: find Z 2x cos(x2 ) dx. This is not a “simple” derivative, but a little thought reveals that it must have come from an application of the chain rule. Multiplied on the “outside” is 2x, which is the derivative of the “inside” function x2 . Checking: d d sin(x2 ) = cos(x2 ) x2 = 2x cos(x2 ), dx dx so Z 2x cos(x2 ) dx = sin(x2 ) + C. Even when the chain rule has “produced” a certain derivative, it is not always easy to see. Consider this problem: Z p x3 1 − x2 dx. p There are two factors in this expression, x3 and 1 − x2 , but it is not apparent that the chain rule is involved. Some clever rearrangement reveals that it is: Z x 3 p 1− x2 dx = Z   p 1 (−2x) − (1 − (1 − x2 )) 1 − x2 dx. 2 This looks messy, but we do now have something that looks like the result of the chain √ rule: the function 1 − x2 has been substituted into −(1/2)(1 − x) x, and the derivative 8.1 Substitution 165 of 1 − x2 , −2x, multiplied on the outside. If we can find a function F (x) whose derivative √ is −(1/2)(1 − x) x we’ll be done, since then   p 1 d 2 ′ 2 (1 − (1 − x2 )) 1 − x2 F (1 − x ) = −2xF (1 − x ) = (−2x) − dx 2 p = x3 1 − x2 But this isn’t hard: √ 1 − (1 − x) x dx = 2 1 − (x1/2 − x3/2 ) dx 2   1 2 3/2 2 5/2 +C =− x − x 2 3 5   1 1 x3/2 + C. x− = 5 3 Z Z (8.1.1) So finally we have Z   p 1 1 2 2 x 1 − x dx = (1 − x2 )3/2 + C. (1 − x ) − 5 3 3 So we succeeded, but it required a clever first step, rewriting the original function so that it looked like the result of using the chain rule. Fortunately, there is a technique that makes such problems simpler, without requiring cleverness to rewrite a function in just the right way. It sometimes does not work, or may require more than one attempt, but the idea is simple: guess at the most likely candidate for the “inside function”, then do some algebra to see what this requires the rest of the function to look like. One frequently good guess is any complicated expression inside a square root, so we start by trying u = 1 − x2 , using a new variable, u, for convenience in the manipulations that follow. Now we know that the chain rule will multiply by the derivative of this inner function: du = −2x, dx so we need to rewrite the original function to include this: Z x 3 p 1 − x2 = Z √ −2x x u dx = −2x 3 Z x2 √ du dx. u −2 dx Recall that one benefit of the Leibniz notation is that it often turns out that what looks like ordinary arithmetic gives the correct answer, even if something more complicated is 166 Chapter 8 Techniques of Integration going on. For example, in Leibniz notation the chain rule is dy dt dy = . dx dt dx The same is true of our current expression: Z x2 √ du u dx = −2 dx Z x2 √ u du. −2 Now we’re almost there: since u = 1 − x2 , x2 = 1 − u and the integral is Z √ 1 − (1 − u) u du. 2 It’s no coincidence that this is exactly the integral we computed in (8.1.1), we have simply renamed the variable u to make the calculations less confusing. Just as before:   Z √ 1 1 1 u3/2 + C. u− − (1 − u) u du = 2 5 3 Then since u = 1 − x2 :   Z p 1 1 3 2 x 1 − x2 dx = (1 − x2 )3/2 + C. (1 − x ) − 5 3 To summarize: if we suspect that a given function is the derivative of another via the chain rule, we let u denote a likely candidate for the inner function, then translate the given function so that it is written entirely in terms of u, with no x remaining in the expression. If we can integrate this new function of u, then the antiderivative of the original function is obtained by replacing u by the equivalent expression in x. Even in simple cases you may prefer to use this mechanical procedure, since it often helps to avoid silly mistakes. For example, consider again this simple problem: Z 2x cos(x2 ) dx. Let u = x2 , then du/dx = 2x or du = 2x dx. Since we have exactly 2x dx in the original integral, we can replace it by du: Z Z 2 2x cos(x ) dx = cos u du = sin u + C = sin(x2 ) + C. This is not the only way to do the algebra, and typically there are many paths to the correct answer. Another possibility, for example, is: Since du/dx = 2x, dx = du/2x, and 8.1 Substitution 167 then the integral becomes Z Z Z du 2 = cos u du. 2x cos(x ) dx = 2x cos u 2x The important thing to remember is that you must eliminate all instances of the original variable x. Z EXAMPLE 8.1.1 Evaluate (ax + b)n dx, assuming that a and b are constants, a 6= 0, and n is a positive integer. We let u = ax + b so du = a dx or dx = du/a. Then Z Z 1 n 1 1 n (ax + b) dx = u du = un+1 + C = (ax + b)n+1 + C. a a(n + 1) a(n + 1) EXAMPLE 8.1.2 Evaluate Z sin(ax + b) dx, assuming that a and b are constants and a 6= 0. Again we let u = ax + b so du = a dx or dx = du/a. Then Z Z 1 1 1 sin(ax + b) dx = sin u du = (− cos u) + C = − cos(ax + b) + C. a a a EXAMPLE 8.1.3 Evaluate Z 4 x sin(x2 ) dx. First we compute the antiderivative, then 2 evaluate the definite integral. Let u = x2 so du = 2x dx or x dx = du/2. Then Z Z 1 1 1 2 x sin(x ) dx = sin u du = (− cos u) + C = − cos(x2 ) + C. 2 2 2 Now Z 2 4 1 x sin(x ) dx = − cos(x2 ) 2 4 2 2 1 1 = − cos(16) + cos(4). 2 2 A somewhat neater alternative to this method is to change the original limits to match the variable u. Since u = x2 , when x = 2, u = 4, and when x = 4, u = 16. So we can do this: Z 4 Z 16 16 1 1 1 1 2 x sin(x ) dx = sin u du = − (cos u) = − cos(16) + cos(4). 2 2 2 2 2 4 4 An incorrect, and dangerous, alternative is something like this: Z 4 Z 4 4 4 1 1 1 1 1 2 2 x sin(x ) dx = sin u du = − cos(u) = − cos(x ) = − cos(16) + cos(4). 2 2 2 2 2 2 2 2 2 Z 4 1 This is incorrect because sin u du means that u takes on values between 2 and 4, which 2 2 1 is wrong. It is dangerous, because it is very easy to get to the point − cos(u) 2 4 and forget 2 168 Chapter 8 Techniques of Integration 1 1 to substitute x2 back in for u, thus getting the incorrect answer − cos(4) + cos(2). A 2 2 somewhat clumsy, but acceptable, alternative is something like this: Z 4 2 x sin(x ) dx = 2 Z x=4 x=2 x=4 1 1 sin u du = − cos(u) 2 2 x=2 4 1 = − cos(x2 ) 2 2 =− cos(16) cos(4) + . 2 2 1/2 cos(πt) dt. Let u = sin(πt) so du = π cos(πt) dt or 2 1/4 sin (πt) √ du/π = cos(πt) dt. We change the limits to sin(π/4) = 2/2 and sin(π/2) = 1. Then EXAMPLE 8.1.4 Z 1/2 1/4 Evaluate cos(πt) dt = sin2 (πt) Z 1 √ 2/2 Z 1 1 du = π u2 Z 1 √ 2/2 1 −2 1 u−1 u du = π π −1 1 √ 2/2 √ 2 1 . =− + π π Exercises 8.1. Find the antiderivatives or evaluate the definite integral in each problem. Z Z 9 1. (1 − t) dt ⇒ 2. (x2 + 1)2 dx ⇒ Z Z 1 2 100 √ dt ⇒ 3. x(x + 1) dx ⇒ 4. 3 1 − 5t Z Z p 3 5. sin x cos x dx ⇒ 6. x 100 − x2 dx ⇒ Z Z   x2 √ dx ⇒ 8. cos(πt) cos sin(πt) dt ⇒ 7. 1 − x3 Z Z sin x 9. dx ⇒ 10. tan x dx ⇒ cos3 x Z π Z 5 11. sin (3x) cos(3x) dx ⇒ 12. sec2 x tan x dx ⇒ 0 13. Z 15. Z 17. Z 19. Z √ π/2 2 2 2 x sec (x ) tan(x ) dx ⇒ 0 4 3 1 dx ⇒ (3x − 7)2 6x dx ⇒ (x2 − 7)1/9 14. Z π/6 16. Z 1 18. Z 20. Z 0 −1 1 −1 sin7 x dx ⇒ sin(tan x) dx ⇒ cos2 x (cos2 x − sin2 x) dx ⇒ (2x3 − 1)(x4 − 2x)6 dx ⇒ f (x)f ′ (x) dx ⇒ 8.2 8.2 Powers of sine and Powers of sine and cosine 169 osine Functions consisting of products of the sine and cosine can be integrated by using substitution and trigonometric identities. These can sometimes be tedious, but the technique is straightforward. Some examples will suffice to explain the approach. Evaluate EXAMPLE 8.2.1 Z 5 sin x dx = Z Z sin5 x dx. Rewrite the function: 4 sin x sin x dx = Z 2 2 sin x(sin x) dx = Z sin x(1 − cos2 x)2 dx. Now use u = cos x, du = − sin x dx: Z 2 2 sin x(1 − cos x) dx = = Z Z −(1 − u2 )2 du −(1 − 2u2 + u4 ) du 2 1 = −u + u3 − u5 + C 3 5 1 2 = − cos x + cos3 x − cos5 x + C. 3 5 EXAMPLE 8.2.2 Evaluate function: Z 6 sin x dx = Z Z 2 3 sin6 x dx. Use sin2 x = (1 − cos(2x))/2 to rewrite the (1 − cos 2x)3 dx 8 Z 1 = 1 − 3 cos 2x + 3 cos2 2x − cos3 2x dx. 8 Z (sin x) dx = Now we have four integrals to evaluate: Z and Z 1 dx = x 3 −3 cos 2x dx = − sin 2x 2 170 Chapter 8 Techniques of Integration are easy. The cos3 2x integral is like the previous example: Z Z 3 − cos 2x dx = − cos 2x cos2 2x dx Z = − cos 2x(1 − sin2 2x) dx Z 1 = − (1 − u2 ) du 2   u3 1 u− =− 2 3   sin3 2x 1 sin 2x − . =− 2 3 And finally we use another trigonometric identity, cos2 x = (1 + cos(2x))/2:   Z Z 1 + cos 4x sin 4x 3 2 3 cos 2x dx = 3 x+ . dx = 2 2 4 So at long last we get     Z x sin3 2x 3 sin 4x 3 1 6 sin x dx = − sin 2x − + x+ + C. sin 2x − 8 16 16 3 16 4 EXAMPLE 8.2.3 Evaluate Z sin2 x cos2 x dx. Use the formulas sin2 x = (1−cos(2x))/2 and cos2 x = (1 + cos(2x))/2 to get: Z Z 1 − cos(2x) 1 + cos(2x) 2 2 · dx. sin x cos x dx = 2 2 The remainder is left as an exercise. Exercises 8.2. Find Zthe antiderivatives. 1. 3. Z 5. Z 7. Z 9. Z sin2 x dx ⇒ 2. Z sin3 x dx ⇒ sin4 x dx ⇒ 4. Z cos2 x sin3 x dx ⇒ 6. Z sin2 x cos2 x dx ⇒ 8. Z sin x(cos x)3/2 dx ⇒ 10. Z tan3 x sec x dx ⇒ 3 cos x dx ⇒ 3 2 2 2 cos x sin x dx ⇒ sec x csc x dx ⇒ 8.3 8.3 Trigonometri Trigonometric Substitutions 171 Substitutions So far we have seen that it sometimes helps to replace a subexpression of a function by a single variable. Occasionally it can help to replace the original variable by something more complicated. This seems like a “reverse” substitution, but it is really no different in principle than ordinary substitution. EXAMPLE 8.3.1 Evaluate Z p 1 − x2 dx. Let x = sin u so dx = cos u du. Then Z p Z p Z √ 2 2 1 − x dx = 1 − sin u cos u du = cos2 u cos u du. √ We would like to replace cos2 u by cos u, but this is valid only if cos u is positive, since √ cos2 u is positive. Consider again the substitution x = sin u. We could just as well think of this as u = arcsin x. If we do, then by the definition of the arcsine, −π/2 ≤ u ≤ π/2, so cos u ≥ 0. Then we continue: Z √ cos2 u cos u du = = Z 2 cos u du = Z u sin 2u 1 + cos 2u du = + +C 2 2 4 arcsin x sin(2 arcsin x) + + C. 2 4 This is a perfectly good answer, though the term sin(2 arcsin x) is a bit unpleasant. It is possible to simplify this. Using the identity sin 2x = 2 sin x cos x, we can write sin 2u = q p p 2 2 sin u cos u = 2 sin(arcsin x) 1 − sin u = 2x 1 − sin2 (arcsin x) = 2x 1 − x2 . Then the full antiderivative is √ √ arcsin x x 1 − x2 arcsin x 2x 1 − x2 + = + + C. 2 4 2 2 This type of substitution is usually indicated when the function you wish to integrate contains a polynomial expression that might allow you to use the fundamental identity sin2 x + cos2 x = 1 in one of three forms: cos2 x = 1 − sin2 x sec2 x = 1 + tan2 x tan2 x = sec2 x − 1. If your function contains 1 − x2 , as in the example above, try x = sin u; if it contains 1 + x2 try x = tan u; and if it contains x2 − 1, try x = sec u. Sometimes you will need to try something a bit different to handle constants other than one. 172 Chapter 8 Techniques of Integration EXAMPLE 8.3.2 Z p Evaluate 4 − 9x2 dx. We start by rewriting this so that it looks more like the previous example: Z p 4− 9x2 Z p Z p 2 dx = 4(1 − (3x/2) ) dx = 2 1 − (3x/2)2 dx. Now let 3x/2 = sin u so (3/2) dx = cos u du or dx = (2/3) cos u du. Then Z 2 p 1− (3x/2)2 dx = = = = = = Z 4 2 1 − sin u (2/3) cos u du = cos2 u du 3 4u 4 sin 2u + +C 6 12 2 arcsin(3x/2) 2 sin u cos u + +C 3 3 2 arcsin(3x/2) 2 sin(arcsin(3x/2)) cos(arcsin(3x/2)) + +C 3 3 p 2 arcsin(3x/2) 2(3x/2) 1 − (3x/2)2 + +C 3 3 √ 2 arcsin(3x/2) x 4 − 9x2 + + C, 3 2 Z p 2 using some of the work from example 8.3.1. EXAMPLE 8.3.3 Z p 1+ Evaluate x2 Z p 1 + x2 dx. Let x = tan u, dx = sec2 u du, so Z p Z √ 2 2 dx = 1 + tan u sec u du = sec2 u sec2 u du. Since u = arctan(x), −π/2 ≤ u ≤ π/2 and sec u ≥ 0, so Z √ sec2 2 u sec u du = Z √ sec2 u = sec u. Then sec3 u du. In problems of this type, two integrals come up frequently: Z 3 sec u du and Both have relatively nice expressions but they are a bit tricky to discover. Z sec u du. 8.3 First we do R Trigonometric Substitutions sec u du, which we will need to compute Z sec u du = = Z Z sec u Z 173 sec3 u du: sec u + tan u du sec u + tan u sec2 u + sec u tan u du. sec u + tan u Now let w = sec u + tan u, dw = sec u tan u + sec2 u du, exactly the numerator of the function we are integrating. Thus Z Z Z 1 sec2 u + sec u tan u du = dw = ln |w| + C sec u du = sec u + tan u w = ln | sec u + tan u| + C. Now for Z sec3 u du: sec3 u sec3 u sec3 u (tan2 u + 1) sec u sec u = + = + 2 2 2 2 3 2 3 sec u sec u tan u sec u sec u + sec u tan2 u sec u = + + = + . 2 2 2 2 2 3 We already know how to integrate sec u, so we just need the first quotient. This is “simply” a matter of recognizing the product rule in action: Z sec3 u + sec u tan2 u du = sec u tan u. So putting these together we get Z sec u tan u ln | sec u + tan u| + + C, sec3 u du = 2 2 and reverting to the original variable x: Z p sec u tan u ln | sec u + tan u| + +C 1 + x2 dx = 2 2 sec(arctan x) tan(arctan x) ln | sec(arctan x) + tan(arctan x)| + +C 2 2 √ √ ln | 1 + x2 + x| x 1 + x2 + + C, = 2 2 q p using tan(arctan x) = x and sec(arctan x) = 1 + tan2 (arctan x) = 1 + x2 . = 174 Chapter 8 Techniques of Integration Exercises 8.3. Find Zthe antiderivatives. csc x dx ⇒ 1. 3. 5. 7. 9. 11. 13. 2. Z csc3 x dx ⇒ Z p Z p 2 x − 1 dx ⇒ 4. 9 + 4x2 dx ⇒ Z p Z p 2 x 1 − x dx ⇒ 6. x2 1 − x2 dx ⇒ Z Z p 1 √ x2 + 2x dx ⇒ dx ⇒ 8. 1 + x2 Z Z x2 1 √ dx ⇒ 10. dx ⇒ x2 (1 + x2 ) 4 − x2 √ Z Z x x3 √ √ dx ⇒ dx ⇒ 12. 1−x 4x2 − 1 Z p x2 + 1 dx. (Hint: make the substitution x = sinh(u) and then use exercise 6 Compute in section 4.11.) 14. Fix t > 0. The shaded region in the left-hand graph in figure 4.11.2 is bounded by y = x tanh t, y = 0, and x2 − y2 = 1. Prove that twice the area of this region is t, as claimed in section 4.11. 8.4 Integration by Parts We have already seen that recognizing the product rule can be useful, when we noticed that Z sec3 u + sec u tan2 u du = sec u tan u. As with substitution, we do not have to rely on insight or cleverness to discover such antiderivatives; there is a technique that will often help to uncover the product rule. Start with the product rule: d f (x)g(x) = f ′ (x)g(x) + f (x)g ′ (x). dx We can rewrite this as f (x)g(x) dx + Z f (x)g ′ (x) dx, f (x)g (x) dx = f (x)g(x) − Z f ′ (x)g(x) dx. f (x)g(x) = Z ′ and then Z ′ 8.4 Integration by Parts 175 This may not seem particularly useful at first glance, but it turns out that in many cases we have an integral of the form Z f (x)g ′ (x) dx but that Z f ′ (x)g(x) dx is easier. This technique for turning one integral into another is called integration by parts, and is usually written in more compact form. If we let u = f (x) and v = g(x) then du = f ′ (x) dx and dv = g ′ (x) dx and Z u dv = uv − Z v du. To use this technique we need to identify likely candidates for u = f (x) and dv = g ′ (x) dx. EXAMPLE 8.4.1 Z Evaluate let dv = x dx so v = x2 /2 and Z x2 ln x − x ln x dx = 2 EXAMPLE 8.4.2 Z Evaluate dv = sin x dx so v = − cos x and Z x sin x dx = −x cos x − EXAMPLE 8.4.3 Z Z x ln x dx. Let u = ln x so du = 1/x dx. Then we must x2 1 x2 ln x dx = − 2 x 2 Evaluate x x2 ln x x2 dx = − + C. 2 2 4 x sin x dx. Let u = x so du = dx. Then we must let − cos x dx = −x cos x + Z Z Z cos x dx = −x cos x + sin x + C. sec3 x dx. Of course we already know the answer to this, but we needed to be clever to discover it. Here we’ll use the new technique to discover the antiderivative. Let u = sec x and dv = sec2 x dx. Then du = sec x tan x dx and v = tan x 176 Chapter 8 Techniques of Integration and Z 3 sec x dx = sec x tan x − Z tan2 x sec x dx Z (sec2 x − 1) sec x dx Z Z 3 = sec x tan x − sec x dx + sec x dx. = sec x tan x − At first this looks useless—we’re right back to Z Z 3 sec x dx + 2 Z Z Z sec3 x dx. But looking more closely: 3 sec x dx = sec x tan x − 3 sec x dx = sec x tan x + 3 sec x dx = sec x tan x + Z Z 3 sec x dx + Z sec x dx sec x dx Z sec x dx Z Z sec x tan x 1 3 sec x dx = sec x dx + 2 2 = sec x tan x ln | sec x + tan x| + + C. 2 2 Z Evaluate x2 sin x dx. Let u = x2 , dv = sin x dx; then du = 2x dx Z Z 2 2 and v = − cos x. Now x sin x dx = −x cos x + 2x cos x dx. This is better than the EXAMPLE 8.4.4 original integral, but we need to do integration by parts again. Let u = 2x, dv = cos x dx; then du = 2 and v = sin x, and Z 2 2 x sin x dx = −x cos x + Z 2x cos x dx Z 2 = −x cos x + 2x sin x − 2 sin x dx = −x2 cos x + 2x sin x + 2 cos x + C. Such repeated use of integration by parts is fairly common, but it can be a bit tedious to accomplish, and it is easy to make errors, especially sign errors involving the subtraction in the formula. There is a nice tabular method to accomplish the calculation that minimizes the chance for error and speeds up the whole process. We illustrate with the previous example. Here is the table: 8.4 sign u x 2 − 2x − 0 2 Integration by Parts dv u sin x x or − cos x − sin x 2 dv sin x −2x − cos x 0 cos x 2 cos x 177 − sin x To form the first table, we start with u at the top of the second column and repeatedly compute the derivative; starting with dv at the top of the third column, we repeatedly compute the antiderivative. In the first column, we place a “−” in every second row. To form the second table we combine the first and second columns by ignoring the boundary; if you do this by hand, you may simply start with two columns and add a “−” to every second row. To compute with this second table we begin at the top. Multiply the first entry in column u by the second entry in column dv to get −x2 cos x, and add this to the integral of the product of the second entry in column u and second entry in column dv. This gives: 2 −x cos x + Z 2x cos x dx, or exactly the result of the first application of integration by parts. Since this integral is not yet easy, we return to the table. Now we multiply twice on the diagonal, (x2 )(− cos x) and (−2x)(− sin x) and then once straight across, (2)(− sin x), and combine these as 2 −x cos x + 2x sin x − Z 2 sin x dx, giving the same result as the second application of integration by parts. While this integral is easy, we may return yet once more to the table. Now multiply three times on the diagonal to get (x2 )(− cos x), (−2x)(− sin x), and (2)(cos x), and once straight across, (0)(cos x). We combine these as before to get 2 −x cos x + 2x sin x + 2 cos x + Z 0 dx = −x2 cos x + 2x sin x + 2 cos x + C. Typically we would fill in the table one line at a time, until the “straight across” multiplication gives an easy integral. If we can see that the u column will eventually become zero, we can instead fill in the whole table; computing the products as indicated will then give the entire integral, including the “+C ”, as above. 178 Chapter 8 Techniques of Integration Exercises 8.4. Find Zthe antiderivatives. x cos x dx ⇒ 1. 2. Z x2 cos x dx ⇒ 4. Z xex dx ⇒ 6. Z ln x dx ⇒ 8. Z x3 sin x dx ⇒ 3. Z 5. Z 7. Z x arctan x dx ⇒ 9. Z x3 cos x dx ⇒ 10. Z x sin2 x dx ⇒ 11. Z x sin x cos x dx ⇒ 12. Z √ arctan( x) dx ⇒ 13. Z √ sin( x) dx ⇒ 14. Z sec2 x csc2 x dx ⇒ 8.5 x xe dx ⇒ 2 sin x dx ⇒ 2 Rational Fun tions A rational function is a fraction with polynomials in the numerator and denominator. For example, 1 x2 + 1 x3 , , , x2 + x − 6 (x − 3)2 x2 − 1 are all rational functions of x. There is a general technique called “partial fractions” that, in principle, allows us to integrate any rational function. The algebraic steps in the technique are rather cumbersome if the polynomial in the denominator has degree more than 2, and the technique requires that we factor the denominator, something that is not always possible. However, in practice one does not often run across rational functions with high degree polynomials in the denominator for which one has to find the antiderivative function. So we shall explain how to find the antiderivative of a rational function only when the denominator is a quadratic polynomial ax2 + bx + c. We should mention a special type of rational function that we already know how to integrate: If the denominator has the form (ax + b)n , the substitution u = ax + b will always work. The denominator becomes un , and each x in the numerator is replaced by (u − b)/a, and dx = du/a. While it may be tedious to complete the integration if the numerator has high degree, it is merely a matter of algebra. 8.5 Find EXAMPLE 8.5.1 Z 3 1 x dx = (3 − 2x)5 −2 = 1 16 1 = 16 1 = 16 =− Z Z    Z x3 dx. Using the substitution u = 3 − 2x we get (3 − 2x)5 u−3 −2 u5 179 Rational Functions 3 1 du = 16 Z u3 − 9u2 + 27u − 27 du u5 u−2 − 9u−3 + 27u−4 − 27u−5 du u−1 9u−2 27u−3 27u−4 − + − −1 −2 −3 −4  +C (3 − 2x)−1 9(3 − 2x)−2 27(3 − 2x)−3 27(3 − 2x)−4 − + − −1 −2 −3 −4 1 9 9 27 + − + +C 2 3 16(3 − 2x) 32(3 − 2x) 16(3 − 2x) 64(3 − 2x)4  +C We now proceed to the case in which the denominator is a quadratic polynomial. We can always factor out the coefficient of x2 and put it outside the integral, so we can assume that the denominator has the form x2 + bx + c. There are three possible cases, depending on how the quadratic factors: either x2 + bx + c = (x − r)(x − s), x2 + bx + c = (x − r)2 , or it doesn’t factor. We can use the quadratic formula to decide which of these we have, and to factor the quadratic if it is possible. EXAMPLE 8.5.2 Determine whether x2 + x + 1 factors, and factor it if possible. The quadratic formula tells us that x2 + x + 1 = 0 when x= −1 ± √ 2 1−4 . Since there is no square root of −3, this quadratic does not factor. EXAMPLE 8.5.3 Determine whether x2 − x − 1 factors, and factor it if possible. The quadratic formula tells us that x2 − x − 1 = 0 when x= Therefore x2 − x − 1 = 1± √ 1+4 1± 5 = . 2 2 √ √ ! √ ! 1− 5 1+ 5 x− . x− 2 2 180 Chapter 8 Techniques of Integration If x2 + bx + c = (x − r)2 then we have the special case we have already seen, that can be handled with a substitution. The other two cases require different approaches. If x2 + bx + c = (x − r)(x − s), we have an integral of the form Z p(x) dx (x − r)(x − s) where p(x) is a polynomial. The first step is to make sure that p(x) has degree less than 2. x3 dx in terms of an integral with a numerator (x − 2)(x + 3) that has degree less than 2. To do this we use long division of polynomials to discover that EXAMPLE 8.5.4 Rewrite so Z x3 x3 7x − 6 7x − 6 = 2 =x−1+ 2 =x−1+ , (x − 2)(x + 3) x +x−6 x +x−6 (x − 2)(x + 3) Z x3 dx = (x − 2)(x + 3) Z x − 1 dx + Z 7x − 6 dx. (x − 2)(x + 3) The first integral is easy, so only the second requires some work. Now consider the following simple algebra of fractions: B A(x − s) + B(x − r) (A + B)x − As − Br A + = = . x−r x−s (x − r)(x − s) (x − r)(x − s) That is, adding two fractions with constant numerator and denominators (x−r) and (x−s) produces a fraction with denominator (x − r)(x − s) and a polynomial of degree less than 2 for the numerator. We want to reverse this process: starting with a single fraction, we want to write it as a sum of two simpler fractions. An example should make it clear how to proceed. x3 7x − 6 EXAMPLE 8.5.5 Evaluate dx. We start by writing (x − 2)(x + 3) (x − 2)(x + 3) as the sum of two fractions. We want to end up with Z 7x − 6 A B = + . (x − 2)(x + 3) x−2 x+3 If we go ahead and add the fractions on the right hand side we get (A + B)x + 3A − 2B 7x − 6 = . (x − 2)(x + 3) (x − 2)(x + 3) So all we need to do is find A and B so that 7x − 6 = (A + B)x + 3A − 2B, which is to say, we need 7 = A + B and −6 = 3A − 2B. This is a problem you’ve seen before: solve a 8.5 Rational Functions 181 system of two equations in two unknowns. There are many ways to proceed; here’s one: If 7 = A+B then B = 7−A and so −6 = 3A−2B = 3A−2(7−A) = 3A−14+2A = 5A−14. This is easy to solve for A: A = 8/5, and then B = 7 − A = 7 − 8/5 = 27/5. Thus Z Z 7x − 6 27 1 8 27 8 1 dx = + dx = ln |x − 2| + ln |x + 3| + C. (x − 2)(x + 3) 5x−2 5 x+3 5 5 The answer to the original problem is now Z Z Z x3 7x − 6 dx = x − 1 dx + dx (x − 2)(x + 3) (x − 2)(x + 3) = 8 27 x2 − x + ln |x − 2| + ln |x + 3| + C. 2 5 5 Now suppose that x2 + bx + c doesn’t factor. Again we can use long division to ensure that the numerator has degree less than 2, then we complete the square. x+1 dx. The quadratic denominator does not + 4x + 8 factor. We could complete the square and use a trigonometric substitution, but it is simpler to rearrange the integrand: Z Z Z x+2 1 x+1 dx = dx − dx. 2 2 2 x + 4x + 8 x + 4x + 8 x + 4x + 8 EXAMPLE 8.5.6 Evaluate Z x2 The first integral is an easy substitution problem, using u = x2 + 4x + 8: Z Z x+2 1 du 1 dx = = ln |x2 + 4x + 8|. 2 x + 4x + 8 2 u 2 For the second integral we complete the square: ! 2  x + 2 +1 , x2 + 4x + 8 = (x + 2)2 + 4 = 4 2 making the integral 1 4 Using u = Z 1
x+2 2 2 +1 dx. x+2 we get 2   Z Z 1 1 1 2 1 x+2 . dx = du = arctan
x+2 2 4 4 u2 + 1 2 2 + 1 2 The final answer is now   Z 1 1 x+2 x+1 2 + C. dx = ln |x + 4x + 8| − arctan x2 + 4x + 8 2 2 2 182 Chapter 8 Techniques of Integration Exercises 8.5. Find the antiderivatives. Z 1 dx ⇒ 1. 4 − x2 Z 1 3. dx ⇒ x2 + 10x + 25 Z x4 5. dx ⇒ 4 + x2 Z x3 7. dx ⇒ 4 + x2 Z 1 dx ⇒ 9. 2 2x − x − 3 8.6 x4 dx ⇒ 4 − x2 Z x2 4. dx ⇒ 4 − x2 Z 1 6. dx ⇒ 2 x + 10x + 29 Z 1 8. dx ⇒ x2 + 10x + 21 Z 1 10. dx ⇒ 2 x + 3x 2. Z Numeri al Integration We have now seen some of the most generally useful methods for discovering antiderivatives, and there are others. Unfortunately, some functions have no simple antiderivatives; in such cases if the value of a definite integral is needed it will have to be approximated. We will see two methods that work reasonably well and yet are fairly simple; in some cases more sophisticated techniques will be needed. Of course, we already know one way to approximate an integral: if we think of the integral as computing an area, we can add up the areas of some rectangles. While this is quite simple, it is usually the case that a large number of rectangles is needed to get acceptable accuracy. A similar approach is much better: we approximate the area under a curve over a small interval as the area of a trapezoid. In figure 8.6.1 we see an area under a curve approximated by rectangles and by trapezoids; it is apparent that the trapezoids give a substantially better approximation on each subinterval. .... ........ .............. ..... ..... .... ..... . . .... . .... ... . ... . .. ... . ... . . ... . . . ... . . ... . . . ... . ... . ... ... ... ... ... ... . . .... . .... ... .... ... .... ... . . ..... .. ..... .... ..... ..... ...... ..................... Figure 8.6.1 .......... ....... ........... ..... ... .............. ... .... ........ . . .. .. ..... .. ... .... ... ... ... ... ... ...... . ...... . . . . ....... . ... . . ...... . .... . ...... . .... . ...... . .... ..... ... ...... ..... .. .. ....... ...... ..... . . ....... ... . ...... .. ... ....... ... .. .. ..... ....... . . ........ . ... ... ........ ........ ..... ....... ........... ...... .............. Approximating an area with rectangles and with trapezoids. (AP) As with rectangles, we divide the interval into n equal subintervals of length ∆x. A f (xi ) + f (xi+1 ) typical trapezoid is pictured in figure 8.6.2; it has area ∆x. If we add up 2 8.6 Numerical Integration 183 the areas of all trapezoids we get f (x1 ) + f (x2 ) f (xn−1 ) + f (xn ) f (x0 ) + f (x1 ) ∆x + ∆x + · · · + ∆x = 2 2 2   f (x0 ) f (xn ) ∆x. + f (x1 ) + f (x2 ) + · · · + f (xn−1 ) + 2 2 This is usually known as the Trapezoid Rule. For a modest number of subintervals this is not too difficult to do with a calculator; a computer can easily do many subintervals. (xi , f (xi )) ................... ... ..... ... .... ... ..... ... .... ... ... ....... ...... ...... ...... .... xi Figure 8.6.2 (xi+1 , f (xi+1 )) xi+1 A single trapezoid. In practice, an approximation is useful only if we know how accurate it is; for example, we might need a particular value accurate to three decimal places. When we compute a particular approximation to an integral, the error is the difference between the approximation and the true value of the integral. For any approximation technique, we need an error estimate, a value that is guaranteed to be larger than the actual error. If A is an approximation and E is the associated error estimate, then we know that the true value of the integral is between A − E and A + E. In the case of our approximation of the integral, we want E = E(∆x) to be a function of ∆x that gets small rapidly as ∆x gets small. Fortunately, for many functions, there is such an error estimate associated with the trapezoid approximation. THEOREM 8.6.1 Suppose f has a second derivative f ′′ everywhere on the interval [a, b], and |f ′′ (x)| ≤ M for all x in the interval. With ∆x = (b − a)/n, an error estimate for the trapezoid approximation is E(∆x) = Let’s see how we can use this. b−a (b − a)3 M (∆x)2 = M. 12 12n2 184 Chapter 8 Techniques of Integration EXAMPLE 8.6.2 −x2 Approximate 2 −x2 Z 1 2 e−x dx to two decimal places. The second deriva0 2 tive of f = e is (4x −2)e , and it is not hard to see that on [0, 1], |(4x2 −2)e−x | ≤ 2. We begin by estimating the number of subintervals we are likely to need. To get two decimal places of accuracy, we will certainly need E(∆x) < 0.005 or 1 1 (2) 2 < 0.005 12 n 1 (200) < n2 6 r 100