## El_Arrow one year ago please need help with these two integral problems

1. El_Arrow

$\int\limits_{0}^{\pi/2} \sin^2x*\cos^2x$

2. El_Arrow

and $\int\limits_{?}^{?} \sin8x * \cos5x$

3. El_Arrow

@Concentrationalizing @perl @zepdrix @freckles

4. El_Arrow

how do i do them?

5. anonymous

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

why by 4?

7. anonymous

and for the second one 2Sin A cos B = Sin (A+B) + Sin (A-B)

8. freckles

$\sin(x+y)=\sin(x)\cos(y)+\sin(y) \cos(x) \\ \sin(x-y)=\sin(x)\cos(y)-\sin(y)\cos(x) \\ \text{ add these identities } \sin(x+y)+\sin(x-y)=2 \sin(x) \cos(y) +0 \\ \text{ now divide by } 2 \\ \frac{\sin(x+y)+\sin(x-y)}{2}=\sin(x) \cos(y) \\ \frac{1}{2} \sin(x+y)+\frac{1}{2} \sin(x-y)=\sin(x)\cos(y)$ for the second one (Which I'm late for )

9. anonymous

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

okay so i got sin(13)+sin(3) for the second one

11. El_Arrow

so now i divide the whole thing by 2?

12. anonymous

check this exp again sin (13) + sin (3) isn't there something missing?

13. El_Arrow

oh the 1/2's right

14. anonymous

its 'x' dear

15. anonymous

sin (13x) + sin (3x)

16. El_Arrow

x's where do they go?

17. anonymous

got it now?

18. El_Arrow

19. anonymous

no, now you have to integrate them wrt x

20. El_Arrow

how?

21. El_Arrow

oh you mean turn them to cosines

22. anonymous

yeah

23. El_Arrow

okay so it would be 13cos(13x) + 3cos(3x)

24. El_Arrow

right?

25. anonymous

you should see your notes again those 12 and 3 will be in the denom

26. El_Arrow

okay fixed it so for the first one its a trig identity right?

27. anonymous

yeah

28. El_Arrow

so when its sin^2x*cos^2x you always change it to 4sin^2x*cos^2x?

29. anonymous

yep

30. El_Arrow

okay

31. El_Arrow

okay so i got |dw:1433738264454:dw|

32. El_Arrow

whats the next step @divu.mkr

33. El_Arrow

@freckles do you know how to do the first one?

34. freckles

$\int\limits _0^\frac{\pi}{2}\sin^2(x) \cos^2(x) dx \\ \int\limits_0^\frac{\pi}{2} \frac{1}{4} 4 \sin^2(x) \cos^2(x) dx \\ \frac{1}{4} \int\limits_0^\frac{\pi}{2} (2 \sin(x) \cos(x))^2 dx \\ \frac{1}{4} \int\limits_0^\frac{1}{2}( \sin(2x))^2 dx \\$ 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

$\frac{1}{4}\int\limits_0^\frac{1}{2} \frac{1}{2}(1-\cos(2[2x])) dx$

36. El_Arrow

my god its a lot to memorize

37. freckles

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

so when did you use u-substitution?

39. freckles

I didn't yet

40. El_Arrow

cause you changed the pi/2 to 1/2

41. freckles

you could those on the 4x thing inside the cos thing

42. freckles

oh that was a type-o

43. El_Arrow

oh okay so how many trig identities should i know

44. El_Arrow

i mean just identities not trig

45. freckles

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

there is also methods that are useful to depending on what you are doing

47. freckles

like you pretty much want to remember everything you can from your algebra and trig days

48. freckles

calculus pretty much depends on all of that

49. El_Arrow

yeah this is why i hate calculus

50. freckles

@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

Inspirational words needed

52. El_Arrow

i just hated it cause its stressful sometimes you know

53. El_Arrow

but i do enjoy working out problems and stuff

54. freckles

maybe it is just overwhelming I guess

55. freckles

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

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

guess it just takes some time to decouple calculus from algebra and see its beauty

58. El_Arrow

i remember some stuff from high school math like the trig identities and pythagoeron theorem and other stuff

59. El_Arrow

yeah i guess your right just takes some time you know

60. freckles

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

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

eventually though but long after calculus 1 I was able to recall more trig identities

63. ganeshie8

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

i think that if i had taken calculus in high school i wouldnt be struggling so much

65. El_Arrow

what about for calculus 2 and 3?

66. ganeshie8

you wont be struggling as much if you memorize those few trig identities

67. El_Arrow

68. ganeshie8

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

i hope so

70. ganeshie8

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

oh okay thank you professors/authors

72. El_Arrow

well its been nice chatting with you guys but i get wake up early tomorrow for calculus class

73. El_Arrow

good night and thanks for the advise

74. freckles

@ganeshie8 is so awesome that way

75. ganeshie8

gnite have horrible calc dreams ;p

76. freckles

i used to have horrible math dreams

77. freckles

for real

78. El_Arrow

lol thanks

79. freckles

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

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

@ganeshie8 It was never relaxing for me to sleep back then.

82. freckles

I couldn't get the math out of my head.

83. ganeshie8

there is some theory that you relax more when you dream

84. freckles

It is all lies.

85. ganeshie8

something like... you dream only when you're in deep stage of sleep

86. freckles

I wonder if I was really sleep all of those times then.