Integration by substitution There are occasions when it is possible to perform an apparen tly dicult piece of integration by rst making a substitution
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Presentation on theme: "Integration by substitution There are occasions when it is possible to perform an apparen tly dicult piece of integration by rst making a substitution"— Presentation transcript:
Page 1 Integration by substitution There are occasions when it is possible to perform an apparen tly diﬃcult piece of integration by ﬁrst making a substitution . This has the eﬀect of changing the variable and the integran d. When dealing with deﬁnite integrals, the limits of integrat ion can also change. In this unit we will meet several examples of integrals where it is appropri ate to make a substitution. In order to master the techniques explained here it is vital t hat you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial o n this topic, you should be able to: carry out integration by making a substitution identify appropriate substitutions to make in order to eval uate an integral Contents 1. Introduction 2. Integration by substituting ax 3. Finding )) ) d by substituting ) 6 math centre December 1, 2008 Page 2 1. Introduction There are occasions when it is possible to perform an apparen tly diﬃcult piece of integration by ﬁrst making a substitution . This has the eﬀect of changing the variable and the integran d. When dealing with deﬁnite integrals, the limits of integrat ion can also change. In this unit we will meet several examples of this type. The ability to carry out integration by substitution is a skill that develops with practice and experience. For this r eason you should carry out all of the practice exercises. Be aware that sometimes an apparently s ensible substitution does not lead to an integral you will be able to evaluate. You must then be pr epared to try out alternative substitutions. 2. Integration by substituting ax We introduce the technique through some simple examples for which a linear substitution is appropriate. Example Suppose we want to ﬁnd the integral + 4) (1) You will be familiar already with ﬁnding a similar integral and know that this integral is equal to , where is a constant of integration. This is because you know that th e rule for integrating powers of a variable tells you to increase th e power by 1 and then divide by the new power. In the integral given by Equation (1) there is still a power 5, but the integrand is more compli- cated due to the presence of the term + 4. To tackle this problem we make a substitution We let + 4. The point of doing this is to change the integrand into the much simpler However, we must take care to substitute appropriately for t he term d too. In terms of diﬀerentials we have Now, in this example, because + 4 it follows immediately that = 1 and so d = d So, substituting both for + 4 and for d in Equation (1) we have + 4) The resulting integral can be evaluated immediately to give . We can revert to an expression involving the original variable by recalling that + 4, giving + 4) + 4) We have completed the integration by substitution. math centre December 1, 2008 Page 3 Example Suppose now we wish to ﬁnd the integral cos(3 + 4) d (2) Observe that if we make a substitution = 3 + 4, the integrand will then contain the much simpler form cos which we will be able to integrate. As before, and so with = 3 + 4 and = 3 It follows that = 3 d So, substituting for 3 + 4, and with d in Equation (2) we have cos(3 + 4) d cos sin We can revert to an expression involving the original variab le by recalling that = 3 + 4, giving cos(3 + 4) d sin(3 + 4) + We have completed the integration by substitution. It is very easy to generalise the result of the previous examp le. If we want to ﬁnd cos( ax )d the substitution ax leads to cos which equals sin , that is sin( ax )+ A similar argument, which you should try, shows that sin( ax )d cos( ax ) + Key Point sin( ax )d cos( ax ) + cos( ax )d sin( ax ) + math centre December 1, 2008 Page 4 Example Suppose we wish to ﬁnd We make the substitution = 1 in order to simplify the integrand to . Recall that the integral of with respect to is the natural logarithm of , ln . As before, and so with = 1 and It follows that 2 d The integral becomes ln ln The result of the previous example can be generalised: if we w ant to ﬁnd ax , the substitution ax leads to which equals ln ax This means, for example, that when faced with an integral suc h as + 7 we can imme- diately write down the answer as ln + 7 Key Point ax ln ax math centre December 1, 2008 Page 5 A little more care must be taken with the limits of integratio n when dealing with deﬁnite integrals. Consider the following example. Example Suppose we wish to ﬁnd (9 + We make the substitution = 9 + . As before, and so with = 9 + and = 1 It follows that = d The integral becomes =3 =1 where we have explicitly written the variable in the limits o f integration to emphasise that those limits were on the variable and not . We can write these as limits on using the substitution = 9 + . Clearly, when = 1, = 10, and when = 3, = 12. So we require =12 =10 12 10 12 10 728 Note that in this example there is no need to convert the answe r given in terms of back into one in terms of because we had already converted the limits on into limits on Exercises 1. 1. In each case use a substitution to ﬁnd the integral: (a) 2) (b) + 5) (c) (2 1) (d) (1 2. In each case use a substitution to ﬁnd the integral: (a) sin(7 3)d (b) (c) π/ cos(1 )d (d) + 5 math centre December 1, 2008 Page 6 3. Finding )) ) d by substituting Example Suppose now we wish to ﬁnd the integral 1 + (3) In this example we make the substitution = 1 + , in order to simplify the square-root term. We shall see that the rest of the integrand, 2 , will be taken care of automatically in the substitution process, and that this is because 2 is the derivative of that part of the integrand used in the substitution, i.e. 1 + As before, and so with = 1 + and = 2 It follows that = 2 So, substituting for 1 + , and with 2 = d in Equation (3) we have 1 + We can revert to an expression involving the original variab le by recalling that = 1 + giving 1 + (1 + We have completed the integration by substitution. Let us analyse this example a little further by comparing the integrand with the general case )) ). Suppose we write ) = 1 + and ) = Then we note that the composition of the functions and is )) = 1 + when ﬁnding the composition of functions and it is the output from which is used as input to resulting in )) math centre December 1, 2008 Page 7 Further, we note that if ) = 1 + then ) = 2 . So the integral 1 + is of the form )) ) d To perform the integration we used the substitution = 1 + . In the general case it will be appropriate to try substituting ). Then d )d Once the substitution was made the resulting integral becam . In the general case it will become ) d . Provided that this ﬁnal integral can be found the problem is solved. For purposes of comparison the speciﬁc example and the gener al case are presented side-by-side: 1 + )) )d let = 1 + let = 2 ) d 1 + )) )d ) d (1 + Key Point To evaluate )) )d substitute ), and d )d to give ) d Integration is then carried out with respect to , before reverting to the original variable It is worth pointing out that integration by substitution is something of an art - and your skill at doing it will improve with practice. Furthermore, a subst itution which at ﬁrst sight might seem sensible, can lead nowhere. For example, if you were try to ﬁnd 1 + by letting = 1 + you would ﬁnd yourself up a blind alley. Be prepared to persev ere and try diﬀerent approaches. math centre December 1, 2008 Page 8 Example Suppose we wish to evaluate + 1 By writing the integrand as + 1 we note that it takes the form )) )d where ) = ) = 2 + 1 and ) = 4 The substitution ) = 2 + 1 transforms the integral to ) d This is evaluated to give = 2 Finally, using = 2 + 1 to revert to the original variable gives + 1 = 2(2 + 1) or equivalently + 1 + Example Suppose we wish to ﬁnd sin Consider the substitution . Then so that sin = 2 sin from which sin 2 cos 2 cos We can also make the following observations: the integrand can be written in the form sin math centre December 1, 2008 Page 9 Writing ) = sin and ) = then ) = Further, )) = sin Hence we write the given integral as sin which is of the form )) ) d with and as given above. As before the substitution ) = produces the integral ) d = 2 sin from which sin 2 cos 2 cos Exercises 2 1. In each case the integrand can be written as )) ). Identify the functions and and use the general result on page 7 to complete the integrati on. (a) (b) sin(1 )d (c) cos 1 + sin 2. In each case use the given substitution to ﬁnd the integral (a) (b) sin(2 )d = 2 (c) + 1d + 1. 3. In each case use a suitable substitution to ﬁnd the integra l. (a) (b) (1 + (c) (1 + (d) + 16 (e) cos (5 + sin (f) + 12 (g) (h) cos sin (i) sin cos math centre December 1, 2008 Page 10 Answers to Exercises Exercises 1 1. (a) 2) (b) 4651 = 930 (c) (2 1) 16 (d) 4. 2. (a) cos(7 3) (b) (c) 1.382 (3dp) (d) ln + 5 Exercises 2 1. (a) ) = e ) = 5, e (b) ) = sin ) = 1 cos(1 ) + (c) ) = ) = 1 + sin , ln 1 + sin 2. (a) e +c (b) cos(2 (c) 2610 (4sf). 3. (a) (1 (b) 1 + (c) 20 (1 + (d) + 16) (e) 5 + sin (f) 0.0707 (g) 10 (1 (h) cos (i) e sin math centre December 1, 2008 10