Section	5.5
 Integration	by	Substitution, Part	Deux

                   V63.0121, Calculus	I



                      Apri...
Yes, there	is	class	on	Monday




      No	new	material
      We	will	review	the	course
      We	will	answer	questions, so...
Final	stuff

               Old	finals	online, including	Fall	2008
               Review	sessions: May	5	and	6, 6:00–8:00pm...
Resurrection	Policy
        If	your	final	score	beats	your	midterm	score, we	will	add	10%	to
        its	weight, and	subtra...
Outline


  Recall: The	method	of	substitution


  Multiple	substitutions


  Odd	and	even	functions
    Examples


  More...
Last	Time: The	Substitution	Rule



   Theorem
   If u = g(x) is	a	differentiable	function	whose	range	is	an	interval I
  ...
Last	Time: The	Substitution	Rule	for	Definite	Integrals


   Theorem
   If g′ is	continuous	and f is	continuous	on	the	rang...
Last	Time: The	Substitution	Rule	for	Definite	Integrals


   Theorem
   If g′ is	continuous	and f is	continuous	on	the	rang...
Outline


  Recall: The	method	of	substitution


  Multiple	substitutions


  Odd	and	even	functions
    Examples


  More...
An	exponential	example
  Example√
         ∫                      √
             ln       8
                          e2x ...
An	exponential	example
  Example√
         ∫                      √
             ln       8
                          e2x ...
An	exponential	example
  Example√
         ∫                      √
             ln       8
                          e2x ...
Another	way	to	skin	that	cat

   Example√
          ∫                      √
              ln       8
                    ...
Another	way	to	skin	that	cat

   Example√
          ∫                      √
              ln       8
                    ...
Another	way	to	skin	that	cat

   Example√
          ∫                      √
              ln       8
                    ...
Another	way	to	skin	that	cat

   Example√
          ∫                      √
              ln       8
                    ...
Another	way	to	skin	that	cat

   Example√
          ∫                      √
              ln       8
                    ...
A third	skinned	cat


   Example√
          ∫                      √
              ln       8
                           e...
A third	skinned	cat


   Example√
          ∫                      √
              ln       8
                           e...
A third	skinned	cat


   Example√
          ∫                         √
              ln        8
                        ...
Outline


  Recall: The	method	of	substitution


  Multiple	substitutions


  Odd	and	even	functions
    Examples


  More...
Example
    ∫  π
            sin(x) dx
Find
       −π




                        .   .   .   .   .   .
Example
    ∫    π
             sin(x) dx
Find
        −π

Solution

∫   π
        sin(x) = − cos(x)|π = cos(x)|−π = cos(−...
Example
    ∫     π
              sin(x) dx
Find
         −π

Solution

 ∫   π
         sin(x) = − cos(x)|π = cos(x)|−π = ...
Even	and	Odd	Functions


  Definition
  A function f is even if	for	all x,

                                f(−x) = f(x)

 ...
Even	and	Odd	Functions


  Definition
  A function f is even if	for	all x,

                                f(−x) = f(x)

 ...
Even	and	Odd	Functions


  Definition
  A function f is even if	for	all x,

                                f(−x) = f(x)

 ...
Even	and	Odd	Functions


  Definition
  A function f is even if	for	all x,

                                f(−x) = f(x)

 ...
Even	and	Odd	Functions


  Definition
  A function f is even if	for	all x,

                                f(−x) = f(x)

 ...
Even	and	Odd	functions	pictured

                        y
                        .
                 o
                 ....
Examples	of	symmetric	functions




   Even	and	odd	functions	abound.
       x → xn is	odd	when n is	odd	and	even	when n i...
Combining	symmetric	functions



  Theorem
   (a) The	sum	of	even	functions	is	even. The	sum	of	odd	functions
       is	od...
Integrating	symmetric	functions


   Theorem
   Let a be	any	number.
   (a) If f is	odd, then   ∫   a
                    ...
Integrating	symmetric	functions


   Theorem
   Let a be	any	number.
   (a) If f is	odd, then             ∫   a
          ...
Proof	(odd f).
           ∫     a
                     f(x) dx, let u = −x. Then du = −dx and	we	have
To	compute
         ...
Proof	(even f).
With	the	same	substitution	we	have
                 ∫0               ∫                        0
          ...
Example
Compute
          ∫       √
                                   √
              e       π +1

                     ...
Example
Compute
                ∫       √
                                         √
                    e       π +1

   ...
Example
Compute
          ∫
                   (              )
              2
                       x4 + x2 + 3 dx.
   ...
Solution
Because	the	integrand	is	even	we	can	simplify	our	arithmetic. It’s
especially	nice	to	plug	in	zero	since	the	resu...
Outline


  Recall: The	method	of	substitution


  Multiple	substitutions


  Odd	and	even	functions
    Examples


  More...
Example
∫
      x3
             dx
  (5x4 + 2)2




                  .   .   .   .   .   .
Example
∫
      x3
             dx
  (5x4 + 2)2
Solution
Let u = 5x4 + 2, so du = 20x3 dx. Then
             ∫            ...
Example
∫
  sin(sin(θ)) cos(θ) dθ




                          .   .   .   .   .   .
Example
∫
  sin(sin(θ)) cos(θ) dθ

Solution
Let u = sin(θ), so du = cos(θ) dθ. Then
           ∫                         ∫...
Example
∫
  ex + e−x
           dx
  ex − e−x




                .   .   .   .   .   .
Example
∫
  ex + e−x
           dx
  ex − e−x
Solution
The	numerator	is	the	derivative	of	the	denominator! Let
           ...
Example
∫
    3x
         dx
  1 + 9x




              .   .   .   .   .   .
Example
∫
     3x
          dx
   1 + 9x
Solution
Notice 9x = (32 )x = 32x = (3x )2 . So	let u = 3x ,
du = (ln 3) · 3x dx....
Example
     √
∫
  sec2       x
    √            dx
         x




                      .   .   .   .   .   .
Example
     √
∫
  sec2       x
    √            dx
         x
Solution
          √                1
              x, so d...
Example
∫
   dx
  x ln x




           .   .   .   .   .   .
Example
∫
    dx
   x ln x
Solution
                        1
Let u = ln x, so du =     dx. Then
                        x...
What	do	we	substitute?


      Linear	factors (ax + b) are	easy	substitutions: u = ax + b,
      du = a dx
      Look	for ...
Outline


  Recall: The	method	of	substitution


  Multiple	substitutions


  Odd	and	even	functions
    Examples


  More...
Course	Evaluations




      Please	fill	out	CAS and	departmental	evaluations
      CAS goes	to	SILV 909	(need	a	volunteer)...
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Lesson 27: Integration by Substitution, part II (Section 10 version)

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Lesson 27: Integration by Substitution, part II (Section 10 version)

  1. 1. Section 5.5 Integration by Substitution, Part Deux V63.0121, Calculus I April 29, 2009 Announcements Class on Monday will be review . . . . . .
  2. 2. Yes, there is class on Monday No new material We will review the course We will answer questions, so bring some . . . . . .
  3. 3. Final stuff Old finals online, including Fall 2008 Review sessions: May 5 and 6, 6:00–8:00pm, SILV 703 Final is May 8, 2:00–3:50pm in CANT 101/200 . . Image credit: Pragmagraphr . . . . . .
  4. 4. Resurrection Policy If your final score beats your midterm score, we will add 10% to its weight, and subtract 10% from the midterm weight. . . Image credit: Scott Beale / Laughing Squid . . . . . .
  5. 5. Outline Recall: The method of substitution Multiple substitutions Odd and even functions Examples More examples and advice Course Evaluations . . . . . .
  6. 6. Last Time: The Substitution Rule Theorem If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ∫ ∫ ′ f(g(x))g (x) dx = f(u) du or ∫ ∫ du f(u) dx = f(u) du dx . . . . . .
  7. 7. Last Time: The Substitution Rule for Definite Integrals Theorem If g′ is continuous and f is continuous on the range of u = g(x), then ∫ ∫ b g(b) f(g(x))g′ (x) dx = f(u) du. a g(a) . . . . . .
  8. 8. Last Time: The Substitution Rule for Definite Integrals Theorem If g′ is continuous and f is continuous on the range of u = g(x), then ∫ ∫ b g(b) f(g(x))g′ (x) dx = f(u) du. a g(a) The integral on the left happens in “x-land”, so its limits are values of x The integral on the right happens in “u-land”, so its limits need to be values of u To convert x to u, simply apply the substitution u = g(x). . . . . . .
  9. 9. Outline Recall: The method of substitution Multiple substitutions Odd and even functions Examples More examples and advice Course Evaluations . . . . . .
  10. 10. An exponential example Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 . . . . . .
  11. 11. An exponential example Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution Let u = e2x , so du = 2e2x dx. We have √ ∫ ∫ √ 8√ ln 8 1 e2x e2x + 1 dx = u + 1 du √ 2 ln 3 3 . . . . . .
  12. 12. An exponential example Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution Let u = e2x , so du = 2e2x dx. We have √ ∫ ∫ √ 8√ ln 8 1 e2x e2x + 1 dx = u + 1 du √ 2 ln 3 3 Now let y = u + 1, dy = du. So ∫ ∫ ∫ 8√ 9 9 √ 1 1 1 y1/2 dy u + 1 du = y dy = 2 2 2 3 4 4 9 12 1 19 = · y3/2 = (27 − 8) = 23 3 3 4 . . . . . .
  13. 13. Another way to skin that cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution Let u = e2x + 1 . . . . . .
  14. 14. Another way to skin that cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution Let u = e2x + 1, so that du = 2e2x dx. . . . . . .
  15. 15. Another way to skin that cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution Let u = e2x + 1, so that du = 2e2x dx. Then √ ∫ ∫ √ ln 8 9√ 1 2x e2x e + 1 dx = u du √ 2 ln 3 4 . . . . . .
  16. 16. Another way to skin that cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution Let u = e2x + 1, so that du = 2e2x dx. Then √ ∫ ∫ √ ln 8 9√ 1 2x e2x e + 1 dx = u du √ 2 ln 3 4 9 1 3/2 u = 3 4 . . . . . .
  17. 17. Another way to skin that cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution Let u = e2x + 1, so that du = 2e2x dx. Then √ ∫ ∫ √ ln 8 9√ 1 2x e2x e + 1 dx = u du √ 2 ln 3 4 1 3/2 9 u = 3 4 1 19 = (27 − 8) = 3 3 . . . . . .
  18. 18. A third skinned cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution √ e2x + 1, so that Let u = u2 = e2x + 1 . . . . . .
  19. 19. A third skinned cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution √ e2x + 1, so that Let u = u2 = e2x + 1 =⇒ 2u du = 2e2x dx . . . . . .
  20. 20. A third skinned cat Example√ ∫ √ ln 8 e2x e2x + 1 dx Find √ ln 3 Solution √ e2x + 1, so that Let u = u2 = e2x + 1 =⇒ 2u du = 2e2x dx Thus √ ∫ ∫ √ 3 ln 8 3 13 19 2x e2x u · u du = e + 1 dx = u = √ 3 3 ln 3 2 2 . . . . . .
  21. 21. Outline Recall: The method of substitution Multiple substitutions Odd and even functions Examples More examples and advice Course Evaluations . . . . . .
  22. 22. Example ∫ π sin(x) dx Find −π . . . . . .
  23. 23. Example ∫ π sin(x) dx Find −π Solution ∫ π sin(x) = − cos(x)|π = cos(x)|−π = cos(−π) − cos(π) = 0 −π π −π . . . . . .
  24. 24. Example ∫ π sin(x) dx Find −π Solution ∫ π sin(x) = − cos(x)|π = cos(x)|−π = cos(−π) − cos(π) = 0 −π π −π This is obvious from the graph: y . . x . . . . . . .
  25. 25. Even and Odd Functions Definition A function f is even if for all x, f(−x) = f(x) A function f is odd if for all x, f(−x) = −f(x). . . . . . .
  26. 26. Even and Odd Functions Definition A function f is even if for all x, f(−x) = f(x) A function f is odd if for all x, f(−x) = −f(x). . . . . . .
  27. 27. Even and Odd Functions Definition A function f is even if for all x, f(−x) = f(x) A function f is odd if for all x, f(−x) = −f(x). These properties are revealed in the graph. . . . . . .
  28. 28. Even and Odd Functions Definition A function f is even if for all x, f(−x) = f(x) A function f is odd if for all x, f(−x) = −f(x). These properties are revealed in the graph. An odd function has rotational symmetry about the origin. . . . . . .
  29. 29. Even and Odd Functions Definition A function f is even if for all x, f(−x) = f(x) A function f is odd if for all x, f(−x) = −f(x). These properties are revealed in the graph. An odd function has rotational symmetry about the origin. An even function has reflective symmetry in the y-axis . . . . . .
  30. 30. Even and Odd functions pictured y . o . dd . x . y . e . ven x . . . . . . .
  31. 31. Examples of symmetric functions Even and odd functions abound. x → xn is odd when n is odd and even when n is even. Funny, that! sin is odd and cos is even. . . . . . .
  32. 32. Combining symmetric functions Theorem (a) The sum of even functions is even. The sum of odd functions is odd. (b) The product of even functions is even. The product of odd functions is even. The product of an odd function and an even function is an odd function. (c) If g is even, then f ◦ g is even. The composition of two odd functions is odd. The composition of an even function and an odd function is even. . . . . . .
  33. 33. Integrating symmetric functions Theorem Let a be any number. (a) If f is odd, then ∫ a f(x) dx = 0. −a . . . . . .
  34. 34. Integrating symmetric functions Theorem Let a be any number. (a) If f is odd, then ∫ a f(x) dx = 0. −a (b) If f is even, then ∫ ∫ a a f(x) dx = 2 f(x) dx. −a 0 . . . . . .
  35. 35. Proof (odd f). ∫ a f(x) dx, let u = −x. Then du = −dx and we have To compute −a ∫ ∫ −a a f(x) dx = − f(−u) du −a a ∫ −a f(u) du = a ∫ a =− f(u) du. −a The only number which is equal to its own negative is zero. . . . . . .
  36. 36. Proof (even f). With the same substitution we have ∫0 ∫ 0 f(x) dx = − f(−u) du −a a ∫0 =− f(u) du a ∫a f(u) du. = 0 So ∫ ∫ ∫ ∫ a a a 0 f(x) dx = f(x) dx + f(x) dx = 2 f(x) dx. −a −a 0 0 . . . . . .
  37. 37. Example Compute ∫ √ √ e π +1 sin(x) 1 + cos3 (x) dx. √ −e π −1 . . . . . .
  38. 38. Example Compute ∫ √ √ e π +1 sin(x) 1 + cos3 (x) dx. √ −e π −1 Solution The integrand is odd! So the answer is zero. . . . . . .
  39. 39. Example Compute ∫ ( ) 2 x4 + x2 + 3 dx. −2 . . . . . .
  40. 40. Solution Because the integrand is even we can simplify our arithmetic. It’s especially nice to plug in zero since the result is often zero. ∫ ∫ ( ) 2( ) 2 4 2 x4 + x2 + 3 dx x + x + 3 dx = 2 −2 0 [ ]2 x5 x3 =2 + 3x + 5 3 0 [5 ] 23 2 =2 + 3(2) + 5 3 [ ] 32 8 =2 + +6 5 3 2 (32 · 3 + 8 · 5 + 6 · 15) = 15 2 · 226 = 15 . . . . . .
  41. 41. Outline Recall: The method of substitution Multiple substitutions Odd and even functions Examples More examples and advice Course Evaluations . . . . . .
  42. 42. Example ∫ x3 dx (5x4 + 2)2 . . . . . .
  43. 43. Example ∫ x3 dx (5x4 + 2)2 Solution Let u = 5x4 + 2, so du = 20x3 dx. Then ∫ ∫ x3 1 1 dx = du 4 + 2)2 u2 20 (5x 11 =− · +C 20 u 1 =− +C 20(5x4 + 2) . . . . . .
  44. 44. Example ∫ sin(sin(θ)) cos(θ) dθ . . . . . .
  45. 45. Example ∫ sin(sin(θ)) cos(θ) dθ Solution Let u = sin(θ), so du = cos(θ) dθ. Then ∫ ∫ sin(sin(θ)) cos(θ) dθ = sin(u) du = − cos(u) + C = − cos(sin(θ)) + C . . . . . .
  46. 46. Example ∫ ex + e−x dx ex − e−x . . . . . .
  47. 47. Example ∫ ex + e−x dx ex − e−x Solution The numerator is the derivative of the denominator! Let ( ) u = ex − e−x , so du = ex + e−x dx. Then ∫ ∫ ex + e−x 1 dx = du x − e−x e u = ln |u| + C = ln ex − e−x + C . . . . . .
  48. 48. Example ∫ 3x dx 1 + 9x . . . . . .
  49. 49. Example ∫ 3x dx 1 + 9x Solution Notice 9x = (32 )x = 32x = (3x )2 . So let u = 3x , du = (ln 3) · 3x dx. Then ∫ ∫ 3x 1 1 x dx = du 1 + u2 (1 + 9 ) ln 3 1 arctan(u) + C = ln 3 1 arctan(3x ) + C = ln 3 . . . . . .
  50. 50. Example √ ∫ sec2 x √ dx x . . . . . .
  51. 51. Example √ ∫ sec2 x √ dx x Solution √ 1 x, so du = √ du. Then Let u = 2x √ ∫ ∫ sec2 x dx = 2 sec2 (u) du √ x = 2 tan(u) + C (√ ) = 2 tan x + C . . . . . .
  52. 52. Example ∫ dx x ln x . . . . . .
  53. 53. Example ∫ dx x ln x Solution 1 Let u = ln x, so du = dx. Then x ∫ ∫ dx 1 du = x ln x u = ln |u| + C = ln |ln x| + C . . . . . .
  54. 54. What do we substitute? Linear factors (ax + b) are easy substitutions: u = ax + b, du = a dx Look for function/derivative pairs in the integrand, one to make u and one to make du: xn and xn−1 (fudge the coefficient) sine and cosine (fudge the minus sign) ex and ex ax and ax (fudge the coefficient) √ 1 x and √ (fudge the factor of 2) x 1 ln x and x . . . . . .
  55. 55. Outline Recall: The method of substitution Multiple substitutions Odd and even functions Examples More examples and advice Course Evaluations . . . . . .
  56. 56. Course Evaluations Please fill out CAS and departmental evaluations CAS goes to SILV 909 (need a volunteer) departmental goes to WWH 627 (need another volunteer) Thank you for your input! . . . . . .

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