Chapter 5: Integrals

Transcription

1 Chapter 5: Integrals

2 Section 5.5 The Substitution Rule (u-substitution)

3 Sec. 5.5: The Substitution Rule We know how to find the derivative of any combination of functions Sum rule Difference rule Constant multiple rule Product rule Quotient rule Chain rule

4 Sec. 5.5: The Substitution Rule We would like to know how to find the antiderivative of any combination of functions, but only a few rules exist Sum rule Difference rule Constant multiple rule Product rule (no such rule) Quotient rule (no such rule) Chain rule (no such rule) This makes antiderivatives more difficult. We can t find the antiderivative of just any function

5 Sec. 5.5: The Substitution Rule What we re going to do is learn integration techniques: develop rules so that if you are in certain very specific situations, you will be able to find the antiderivative The substitution rule (today) Integration by parts (Calc. II) Integration by partial fractions (Calc. II) Trig. substitution (Calc. II)

6 Sec. 5.5: The Substitution Rule U-Substitution: The idea is to reverse the chain rule Remember that whenever you know the derivative of a function, you know an antiderivative of another function Ex: Since sin x 2 = 2x cos x 2, we have 2x cos x 2 dx = sin x 2 + C

7 Sec. 5.5: The Substitution Rule When to use u-substitution (hints): When it works! If the derivative of part of the function you re integrating is another part of the function you re integrating The derivative of part of the function needs to be multiplied by that part The number in front of the derivative doesn t matter Make u the part of the integrand (thing you re integrating) whose derivative is also a part of the thing you re integrating

8 Sec. 5.5: The Substitution Rule What performing a u-substitution: Rewrite the integral so that it only has u s in it Make sure the new integral is simpler than the original integral

9 Sec. 5.5: The Substitution Rule U-Substitution To Find An Indefinite Integral: Let u be the appropriate part of the function Find du, then solve for dx Rewrite the integral in terms of u (at the end of this step only u should appear in the integral) The new u integral should be simpler than the original integral Integrate with respect to u At the end, replace u with what you said u equals at the beginning of the problem because the final answer should have x s in it, not u s. Don t forget the +C

10 Sec. 5.5: The Substitution Rule U-Substitution To Find A Definite Integral: Start off as if you are calculating an indefinite integral. Then there are 2 ways to go 1) x-world u-world x-world plug in the numbers 2) x-world u-world change the limits of integration to u numbers and never go back to x world

11 Look at some problems Sec. 5.5: The Substitution Rule 2x sin x 2 dx u = x 2

12 Look at some problems Sec. 5.5: The Substitution Rule cos x e sin x dx u = sin x

13 Look at some problems Sec. 5.5: The Substitution Rule sin 1 x 1 x 2 dx u = sin 1 x

14 Look at some problems Sec. 5.5: The Substitution Rule (ln x) 2 x dx u = ln x

15 Look at some problems Sec. 5.5: The Substitution Rule 3x 2 2x dx u = 2x 3 + 7

16 Look at some problems Sec. 5.5: The Substitution Rule 18x 3 3x 2 x dx u = 3x 2 x + 4

17 Sec. 5.5: The Substitution Rule Look at some problems x 1 4x 2 dx u = 1 4x 2

18 Look at some problems Sec. 5.5: The Substitution Rule tan x dx u = cos x

19 Look at some problems Sec. 5.5: The Substitution Rule sin 2 x cos x dx u = sin x

20 Look at some problems Sec. 5.5: The Substitution Rule sin ( x x) dx u = x

21 Look at some problems Sec. 5.5: The Substitution Rule sin x sin cos x dx u = cos x

22 Look at some problems Sec. 5.5: The Substitution Rule 5 t sin 5 t dt u = 5 t

23 Look at some problems Sec. 5.5: The Substitution Rule sec 2 x tan 2 x dx u = tan x

24 Look at some problems Sec. 5.5: The Substitution Rule sin 2x 1 + cos 2 x dx u = 1 + cos 2 x

25 Sec. 5.5: The Substitution Rule Look at some problems e 1/x x 2 dx u = 1 x

26 Sec. 5.5: The Substitution Rule Ex 1: Find 2x sin x 2 dx

27 Sec. 5.5: The Substitution Rule Ex 2: Find 3x 2 2x dx 1 1

28 Sec. 5.5: The Substitution Rule Ex 3: Find 18x 3 3x 2 x dx

29 Ex 4: Find Sec. 5.5: The Substitution Rule π 4 π 3 tan x dx

30 Sec. 5.5: The Substitution Rule Other Times You Might Use u-substitution Sometimes you make u a part of the thing you are integrating whose derivative is just a number The integral needs to end up simpler than the original integral But!!! Don t ever make u = x because this does nothing!!!

31 Sec. 5.5: The Substitution Rule Other Times You Might Use u-substitution 3x 7 dx u = 3x 7

32 Sec. 5.5: The Substitution Rule Other Times You Might Use u-substitution e 4 9x dx u = 4 9x

33 Sec. 5.5: The Substitution Rule Ex 5: Find 3x 7 dx

34 Sec. 5.5: The Substitution Rule Ex 6: Find 2 1 e 4 9x dx

35 Sec. 5.5: The Substitution Rule Other Times You Might Use u-substitution In general, if after making a u substitution you end up with a simpler integral that you know how to integrate, then do it that way Make sure you are able to get rid of all x terms

36 Sec. 5.5: The Substitution Rule Other Times You Might Use u-substitution x dx 1 + 2x u = 1 + 2x

37 Sec. 5.5: The Substitution Rule Other Times You Might Use u-substitution x 3 x dx u = x 2 + 1

38 Sec. 5.5: The Substitution Rule Ex 7: Find x 1 + 2x dx

39 Sec. 5.5: The Substitution Rule Ex 8: Find x 3 x dx 1 0

40 Chapter 6: Applications of Integration

41 Section 6.5 Average Value of a Function

42 Sec. 6.5: The Average Value of a Function What does average mean? Story Derive formula on board

43 Sec. 6.5: The Average Value of a Function If f is integrable on [a, b], then its average value is f ave = 1 b a a b f x dx

44 Sec. 6.5: The Average Value of a Function Ex 9: Find the average value of the function g t = t on [1, 3] 3+t2

45 Result: Sec. 6.5: The Average Value of a Function

46 Ex 10: Sec. 6.5: The Average Value of a Function a) Find the average value of the function f(x) = 1 x on [1, 4] b) Find the value of c guaranteed by the Mean Value Theorem for Integrals such that f ave = f(c)