Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut

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1 Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f.

2 Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite Integral) If F is an antierivative of f, then f (x) x = F (x) + c is calle the (general) Inefinite Integral of f, where c is an arbitrary constant.

3 Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite Integral) If F is an antierivative of f, then f (x) x = F (x) + c is calle the (general) Inefinite Integral of f, where c is an arbitrary constant. The inefinite integral of a function represents every possible antierivative, since it has been shown that if two functions have the same erivative on an interval then they iffer by a constant on that interval.

4 Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite Integral) If F is an antierivative of f, then f (x) x = F (x) + c is calle the (general) Inefinite Integral of f, where c is an arbitrary constant. The inefinite integral of a function represents every possible antierivative, since it has been shown that if two functions have the same erivative on an interval then they iffer by a constant on that interval. Terminology: When we write f (x) x, f (x) is referre to as the integran.

5 Basic Integration Formulas As with ifferentiation, there are two types of formulas, formulas for the integrals of specific functions an structural type formulas. Each formula for the erivative of a specific function correspons to a formula for the erivative of an elementary function. The following table lists integration formulas sie by sie with the corresponing ifferentiation formulas.

6 Basic Integration Formulas x n x = x n+1 n + 1 if n 1

7 Basic Integration Formulas x n x = x n+1 n + 1 if n 1 x (x n ) = nx n 1

8 Basic Integration Formulas x n x = x n+1 n + 1 if n 1 x (x n ) = nx n 1 sin x x = cos x + c

9 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c x (x n ) = nx n 1 (cos x) = sin x x

10 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c x (x n ) = nx n 1 (cos x) = sin x x cos x x = sin x + c

11 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c cos x x = sin x + c x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x

12 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c cos x x = sin x + c x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x sec 2 x x = tan x + c

13 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c cos x x = sin x + c sec 2 x x = tan x + c x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x x (tan x) = sec2 x

14 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c cos x x = sin x + c sec 2 x x = tan x + c x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x x (tan x) = sec2 x e x x = e x + c

15 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c cos x x = sin x + c sec 2 x x = tan x + c e x x = e x + c x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x x (tan x) = sec2 x x (ex ) = e x

16 Basic Integration Formulas x n x = x n+1 n + 1 if n 1 sin x x = cos x + c cos x x = sin x + c sec 2 x x = tan x + c e x x = e x + c 1 x = ln x + c x x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x x (tan x) = sec2 x x (ex ) = e x

17 Basic Integration Formulas x n x = x n+1 n + 1 if n 1 sin x x = cos x + c cos x x = sin x + c sec 2 x x = tan x + c e x x = e x + c 1 x = ln x + c x x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x x (tan x) = sec2 x x (ex ) = e x x (ln x) = 1 x

18 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c cos x x = sin x + c sec 2 x x = tan x + c e x x = e x + c 1 x = ln x + c x x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x x (tan x) = sec2 x x (ex ) = e x x (ln x) = 1 x k x = kx + c

19 Basic Integration Formulas x n x = x n+1 if n 1 n + 1 sin x x = cos x + c cos x x = sin x + c sec 2 x x = tan x + c e x x = e x + c 1 x = ln x + c x k x = kx + c x (x n ) = nx n 1 (cos x) = sin x x (sin x) = cos x x x (tan x) = sec2 x x (ex ) = e x x (ln x) = 1 x x (kx) = k

20 Structural Type Formulas We may integrate term-by-term:

21 Structural Type Formulas We may integrate term-by-term: kf (x) x = k f (x) x

22 Structural Type Formulas We may integrate term-by-term: kf (x) x = k f (x) x f (x) ± g(x) x = f (x) x ± g(x) x

23 Structural Type Formulas We may integrate term-by-term: kf (x) x = k f (x) x f (x) ± g(x) x = f (x) x ± g(x) x In plain language, the integral of a constant times a function equals the constant times the erivative of the function an the erivative of a sum or ifference is equal to the sum or ifference of the erivatives.

24 Structural Type Formulas We may integrate term-by-term: kf (x) x = k f (x) x f (x) ± g(x) x = f (x) x ± g(x) x In plain language, the integral of a constant times a function equals the constant times the erivative of the function an the erivative of a sum or ifference is equal to the sum or ifference of the erivatives. These formulas come straight from the corresponing formulas for calculating erivatives an are use the same way.

25 Integrating Iniviual Terms When calculating erivatives of iniviual terms, one nees to recognize whether the term is an elementary function, a prouct, a quotient or a composite function. There is a little bit more art to integration, at least if the term is not the erivative of an elementary function.

26 Integrating Iniviual Terms When calculating erivatives of iniviual terms, one nees to recognize whether the term is an elementary function, a prouct, a quotient or a composite function. There is a little bit more art to integration, at least if the term is not the erivative of an elementary function. Integration is essentially the reverse of ifferentiation, so one might expect formulas for reversing the effects of the Prouct Rule, Quotient Rule an Chain Rule. This is almost the case. There is a formula, calle the Integration By Parts Formula, for reversing the effect of the Prouct Rule an there is a technique, calle Substitution, for reversing the effect of the Chain Rule. There is no specific formula or technique for reversing the effect of the Quotient Rule, but one is not really necessary since the Quotient Rule is reunant.

27 Integration is an Art Integration also becomes an art because not only isn t it always obvious whether one shoul resort to Integration By Parts or the Substitution Technique but the use of the Integration By Parts Formula an the Substitution Technique is not as straightforwar as the use of the Prouct, Quotient or Chain Rule.

28 The Substitution Technique The substitution technique may be ivie into the following steps. Every step but the first is purely mechanical. With a little bit of practice (in other wors, make sure you o the homework problems assigne), you shoul have no more ifficulty carrying out a substitution than you shoul be having by now when you ifferentiate.

29 The Substitution Technique The substitution technique may be ivie into the following steps. Every step but the first is purely mechanical. With a little bit of practice (in other wors, make sure you o the homework problems assigne), you shoul have no more ifficulty carrying out a substitution than you shoul be having by now when you ifferentiate. Note: In the following, we will assume that you are trying to calculate an integral f (x)x. If the ummy variable is calle something other than x, then some of the names you woul use for variables might be ifferent.

30 The Substitution Technique Choose a substitution u = g(x).

31 The Substitution Technique Choose a substitution u = g(x). Some suggestions on how to choose a substitution will be mae later.

32 The Substitution Technique Choose a substitution u = g(x). Some suggestions on how to choose a substitution will be mae later. Calculate the erivative u x = g (x).

33 The Substitution Technique Choose a substitution u = g(x). Some suggestions on how to choose a substitution will be mae later. Calculate the erivative u x = g (x). Treating the erivative as if it were a fraction, solve for x:

34 The Substitution Technique Choose a substitution u = g(x). Some suggestions on how to choose a substitution will be mae later. Calculate the erivative u x = g (x). Treating the erivative as if it were a fraction, solve for x: u x = g (x), u = g (x)x, x = u g (x).

35 The Substitution Technique Choose a substitution u = g(x). Some suggestions on how to choose a substitution will be mae later. Calculate the erivative u x = g (x). Treating the erivative as if it were a fraction, solve for x: u x = g (x), u = g (x)x, x = u g (x). Go back to the original integral an replace g(x) by u an u replace x by g (x).

36 The Substitution Technique Simplify.

37 The Substitution Technique Simplify. Every incience of x shoul cancel out at this step. If not, you will nee to try another substitution. Make sure that you simplify properly this is the easiest step to make mistakes uring.

38 The Substitution Technique Simplify. Every incience of x shoul cancel out at this step. If not, you will nee to try another substitution. Make sure that you simplify properly this is the easiest step to make mistakes uring. Integrate.

39 The Substitution Technique Simplify. Every incience of x shoul cancel out at this step. If not, you will nee to try another substitution. Make sure that you simplify properly this is the easiest step to make mistakes uring. Integrate. Replace u by g(x) in your result.

40 The Substitution Technique Simplify. Every incience of x shoul cancel out at this step. If not, you will nee to try another substitution. Make sure that you simplify properly this is the easiest step to make mistakes uring. Integrate. Replace u by g(x) in your result. Check your answer (of course).

41 Choosing an Appropriate Substitution This is the only non-routine part of carrying out a substitution, but shoul not be at all ifficult for any stuent who has mastere the art of ifferentiation. There are two basic tactics for choosing a substitution. Each will work in the vast majority of cases where a routine substitution is neee. Since neither will work in all cases, you nee to be comfortable with both. Therefore, you shoul try using both methos on the same problem wherever possible. (There are quite a few non-routine substitutions that are use in special situations. They nee to be learne separately.)

42 The First Metho The metho most stuents probably fin easiest to use relies on familiarity with the chain rule for ifferentiation. In orer to ecie on a useful substitution, look at the integran an preten that you are going to calculate its erivative rather than its integral. (You usually on t actually have to write anything own you can usually just visualize the steps.) Look to see if there is any step uring which you woul have to use the chain rule. If so, think of the ecomposition you woul have to make, i.e. the step where you woul write own something like y = f (u), u = g(x). Try the substitution u = g(x).

43 The Secon Metho This metho involves looking at parts of the integran an observing whether the erivative of part of the integran equals some other factor of the integran. If so, u may be substitute for that part. (In eciing, you may ignore constant factors, since they are easy to manipulate aroun.)