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El_Arrow

  • one year ago

please need help with these two integral problems

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  1. El_Arrow
    • one year ago
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    \[\int\limits_{0}^{\pi/2} \sin^2x*\cos^2x\]

  2. El_Arrow
    • one year ago
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    and \[\int\limits_{?}^{?} \sin8x * \cos5x\]

  3. El_Arrow
    • one year ago
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    @Concentrationalizing @perl @zepdrix @freckles

  4. El_Arrow
    • one year ago
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    how do i do them?

  5. anonymous
    • one year ago
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    in the first one just multiply and divide by 4 make it sin^2 2x then convert it using 1-cos 2x = 2sin^2 x it will become a basic integral

  6. El_Arrow
    • one year ago
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    why by 4?

  7. anonymous
    • one year ago
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    and for the second one 2Sin A cos B = Sin (A+B) + Sin (A-B)

  8. freckles
    • one year ago
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    \[\sin(x+y)=\sin(x)\cos(y)+\sin(y) \cos(x) \\ \sin(x-y)=\sin(x)\cos(y)-\sin(y)\cos(x) \\ \text{ add these identities } \sin(x+y)+\sin(x-y)=2 \sin(x) \cos(y) +0 \\ \text{ now divide by } 2 \\ \frac{\sin(x+y)+\sin(x-y)}{2}=\sin(x) \cos(y) \\ \frac{1}{2} \sin(x+y)+\frac{1}{2} \sin(x-y)=\sin(x)\cos(y)\] for the second one (Which I'm late for )

  9. anonymous
    • one year ago
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    coz sin 2x = 2 sin x cos x and sin ^2 2x = 4sin^2 x cos^2 x this will replace those two terms

  10. El_Arrow
    • one year ago
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    okay so i got sin(13)+sin(3) for the second one

  11. El_Arrow
    • one year ago
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    so now i divide the whole thing by 2?

  12. anonymous
    • one year ago
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    check this exp again sin (13) + sin (3) isn't there something missing?

  13. El_Arrow
    • one year ago
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    oh the 1/2's right

  14. anonymous
    • one year ago
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    its 'x' dear

  15. anonymous
    • one year ago
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    sin (13x) + sin (3x)

  16. El_Arrow
    • one year ago
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    x's where do they go?

  17. anonymous
    • one year ago
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    got it now?

  18. El_Arrow
    • one year ago
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    so thats the final answer?

  19. anonymous
    • one year ago
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    no, now you have to integrate them wrt x

  20. El_Arrow
    • one year ago
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    how?

  21. El_Arrow
    • one year ago
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    oh you mean turn them to cosines

  22. anonymous
    • one year ago
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    yeah

  23. El_Arrow
    • one year ago
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    okay so it would be 13cos(13x) + 3cos(3x)

  24. El_Arrow
    • one year ago
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    right?

  25. anonymous
    • one year ago
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    you should see your notes again those 12 and 3 will be in the denom

  26. El_Arrow
    • one year ago
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    okay fixed it so for the first one its a trig identity right?

  27. anonymous
    • one year ago
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    yeah

  28. El_Arrow
    • one year ago
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    so when its sin^2x*cos^2x you always change it to 4sin^2x*cos^2x?

  29. anonymous
    • one year ago
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    yep

  30. El_Arrow
    • one year ago
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    okay

  31. El_Arrow
    • one year ago
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    okay so i got |dw:1433738264454:dw|

  32. El_Arrow
    • one year ago
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    whats the next step @divu.mkr

  33. El_Arrow
    • one year ago
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    @freckles do you know how to do the first one?

  34. freckles
    • one year ago
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    \[\int\limits _0^\frac{\pi}{2}\sin^2(x) \cos^2(x) dx \\ \int\limits_0^\frac{\pi}{2} \frac{1}{4} 4 \sin^2(x) \cos^2(x) dx \\ \frac{1}{4} \int\limits_0^\frac{\pi}{2} (2 \sin(x) \cos(x))^2 dx \\ \frac{1}{4} \int\limits_0^\frac{1}{2}( \sin(2x))^2 dx \\ \] there is an awesome idendity here you should remember for this sin things with even powers or cos with even power \[\sin^2(u)=\frac{1}{2}(1-\cos(2u)) \\ \text{ and the other one \to remember } \\ \cos^2(u)=\frac{1}{2}(1+\cos(2u))\]

  35. freckles
    • one year ago
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    \[\frac{1}{4}\int\limits_0^\frac{1}{2} \frac{1}{2}(1-\cos(2[2x])) dx\]

  36. El_Arrow
    • one year ago
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    my god its a lot to memorize

  37. freckles
    • one year ago
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    I won't lie remembering a lot of formulas actually does kind of help :p and cuts down on time to derive other formulas(identities)

  38. El_Arrow
    • one year ago
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    so when did you use u-substitution?

  39. freckles
    • one year ago
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    I didn't yet

  40. El_Arrow
    • one year ago
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    cause you changed the pi/2 to 1/2

  41. freckles
    • one year ago
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    you could those on the 4x thing inside the cos thing

  42. freckles
    • one year ago
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    oh that was a type-o

  43. El_Arrow
    • one year ago
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    oh okay so how many trig identities should i know

  44. El_Arrow
    • one year ago
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    i mean just identities not trig

  45. freckles
    • one year ago
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    that is hard to say http://www.purplemath.com/modules/idents.htm this page has a lot of useful trig identities other identities ... well I remember pascal's triangle to help expand binomials to some positive power

  46. freckles
    • one year ago
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    there is also methods that are useful to depending on what you are doing

  47. freckles
    • one year ago
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    like you pretty much want to remember everything you can from your algebra and trig days

  48. freckles
    • one year ago
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    calculus pretty much depends on all of that

  49. El_Arrow
    • one year ago
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    yeah this is why i hate calculus

  50. freckles
    • one year ago
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    @ganeshie8 I think I seen a post you made recently about people hating math because of something or whatever is there anything you can say here to inspire @El_Arrow into loving calculus ? :p

  51. freckles
    • one year ago
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    Inspirational words needed

  52. El_Arrow
    • one year ago
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    i just hated it cause its stressful sometimes you know

  53. El_Arrow
    • one year ago
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    but i do enjoy working out problems and stuff

  54. freckles
    • one year ago
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    maybe it is just overwhelming I guess

  55. freckles
    • one year ago
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    you get to calculus and maybe forget a lot so you are trying to learn the calculus like things and trying to relearn everything from algebra to trig

  56. ganeshie8
    • one year ago
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    I think calculus1 is simple and easy, it deals mainly with two problems : 1) slope of tangent line 2) area under curve Most of the times, it is actually the algebra/trig/geometry part (which is not calculus) that is painful and makes calculus look messy...

  57. ganeshie8
    • one year ago
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    guess it just takes some time to decouple calculus from algebra and see its beauty

  58. El_Arrow
    • one year ago
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    i remember some stuff from high school math like the trig identities and pythagoeron theorem and other stuff

  59. El_Arrow
    • one year ago
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    yeah i guess your right just takes some time you know

  60. freckles
    • one year ago
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    I think I remember learning some algebra while in calculus. I think I was also still learning algebra even after finishing calculus 1. Algebra is pretty crazy stuff.

  61. freckles
    • one year ago
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    For trig, I think I remember a few trig identities and derive the rest from what I knew when I needed those identities

  62. freckles
    • one year ago
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    eventually though but long after calculus 1 I was able to recall more trig identities

  63. ganeshie8
    • one year ago
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    for calculus1 you only need to memorize below trig identities 1) double angle identities (sin(2x), cos(2x) and tan(2x)) 2) angle sum identities (sin(a+b), cos(a+b), tan(a+b)) apart from the well known pythagorean identity sin^2x+cos^2x=1

  64. El_Arrow
    • one year ago
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    i think that if i had taken calculus in high school i wouldnt be struggling so much

  65. El_Arrow
    • one year ago
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    what about for calculus 2 and 3?

  66. ganeshie8
    • one year ago
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    you wont be struggling as much if you memorize those few trig identities

  67. El_Arrow
    • one year ago
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    what about in differential equations?

  68. ganeshie8
    • one year ago
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    i bet you felt differentiation part of calc1 easy because you know how to differentiat almost anything using product rule, quotient rule and chain rule. Integration requires "guessing", so it confuses everyone in the start... but you will get used to it after working few problems

  69. El_Arrow
    • one year ago
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    i hope so

  70. ganeshie8
    • one year ago
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    `double angle` and `angle sum` are the only trig identities you will be using in most of your calc1,2,3 problems... the focus is about learning calculus, not mastering trigonometry... so professors/authors avoid problems that deal with complicated trig identities in academic courses

  71. El_Arrow
    • one year ago
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    oh okay thank you professors/authors

  72. El_Arrow
    • one year ago
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    well its been nice chatting with you guys but i get wake up early tomorrow for calculus class

  73. El_Arrow
    • one year ago
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    good night and thanks for the advise

  74. freckles
    • one year ago
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    @ganeshie8 is so awesome that way

  75. ganeshie8
    • one year ago
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    gnite have horrible calc dreams ;p

  76. freckles
    • one year ago
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    i used to have horrible math dreams

  77. freckles
    • one year ago
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    for real

  78. El_Arrow
    • one year ago
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    lol thanks

  79. freckles
    • one year ago
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    Like I would be dreaming about trying to solve a math problem and I would be unable to do it in my dream. I would do number crunching in my sleep. Numbers everywhere in my dreams. Math can be pretty haunting.

  80. ganeshie8
    • one year ago
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    lol that sounds real scary... of course there is a slight possibility that you solve one of those open hard problems in sleep and remember after waking up ;) this reminds me of benzene structure discovery :P

  81. freckles
    • one year ago
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    @ganeshie8 It was never relaxing for me to sleep back then.

  82. freckles
    • one year ago
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    I couldn't get the math out of my head.

  83. ganeshie8
    • one year ago
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    there is some theory that you relax more when you dream

  84. freckles
    • one year ago
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    It is all lies.

  85. ganeshie8
    • one year ago
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    something like... you dream only when you're in deep stage of sleep

  86. freckles
    • one year ago
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    I wonder if I was really sleep all of those times then.

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