## rld613 3 years ago James?

1. neverforgetvivistee

give me edals

2. JamesJ

3. rld613

oh that was annoying i cldnt type

4. neverforgetvivistee

give me medals free medals for everyone

5. rld613

can you write the equation again please :D

6. JamesJ

$\frac{d \ }{dx} \int_{17}^x \sin(t^2)/t \ dt$

7. rld613

d/dx(sinx^2/x) - d/dx(sin(17)/17)

8. JamesJ

No

9. rld613

where did i go wrong?

10. JamesJ

look at the earlier example with ln

11. rld613

k i will

12. rld613

wasn't it d/dx(F(x)-F(17))

13. rld613

oh i see where i went wrong!!!!

14. JamesJ

Yes, but F is not the integrand.

15. rld613

F(x)=-cosx/x???

16. JamesJ

Nooo

17. rld613

K just tell me the answer

18. JamesJ

Look at the ln example again.

19. rld613

In the example we subsituted t for x

20. rld613

In the example we subsituted t for x

21. JamesJ

$\frac{d \ }{dx} \int_2^x \ln(t^2+1) \ dt$

22. JamesJ

What is the value of that expression?

23. rld613

i dont know i am feeling very stupid!!!!!! dF/dx=ln(x^2+1)

24. JamesJ

If $F(t) = \int \ln(t^2+1) \ dt$ then $\int_2^x \ln(t^2+1) \ dt = F(x) - F(2)$. Hence $\frac{d \ }{dx} \int_2^x \ln(t^2+1) \ dt = \frac{d \ }{dx} (F(x) - F(2))$

25. rld613

righ

26. JamesJ

and by the Fund Theorem of Calculus, dF/dx = ln(x^2 + 1). Also (d/dx)F(2) = 0 because F(2) is a number, a constant. Hence the entire expression is just ln(x^2+1)

27. missyfredtom

James do you think you help me please?

28. JamesJ

So, return now to the next example where the integrand has a sin function. What is the value of that derivative?

29. JamesJ

@mft, I'll try and get to it. unfortunately you're not the first to ask.

30. rld613

dont take james away form me:(

31. missyfredtom

i can wait

32. JamesJ

quick rld613. I want to move on with you to yet another example afterwards

33. Hero

rld said she didn't need OS anymore

34. JamesJ

so (d/dx) int_17^x sin(t^2)/t dt = ...

35. rld613

i didnt get that why is 17 to th epower of x

36. JamesJ

$(d/dx) \int_{17}^x \sin(t^2)/t \ dt$

37. rld613

yes

38. JamesJ

evaluate. What is the that equal to.

39. rld613

y is it wrong what i showed u b4?

40. rld613

plz be nice to me and show me all the step? :D

41. JamesJ

No. You write out the steps, following line by the line the example we just used for ln. I'm going to help someone else for a while and expect the right answer when I get back ;-)

42. rld613

LOL see u later :D

43. JamesJ

I'm serious. And I'll be back in 5 minutes, so hurry.

44. JamesJ

??

45. rld613

u did u come back already!!!

46. JamesJ

47. rld613

K can u walk me thru the steps cuz like i bever learnt this stuff? YOu r so scary LOL

48. rld613

I was hoping i wld have the answer b4 u wld return

49. rld613

50. JamesJ

Exactly as before, exactly, let $F(t) = \int \sin(t^2)/t \ dt$ Then $\int_{17}^x \sin(t^2)/t \ dt = F(x) - F(17)$ Thus the derivative of this definite integral $\frac{d\ }{dx} (F(x) - F(17)) = ...$

51. rld613

iSo what is F(x)?????

52. rld613

That is where i am going wrong

53. rld613

oh i know si(x)?

54. rld613

d/dx(si(x)?

55. JamesJ

like we've seen, we actually don't need an explicit formula for F(x). We just need to know it is this integral and then use the Fundamental Theorem of Calculus.

56. JamesJ

What is dF/dx here? What is it equal to?

57. JamesJ

By the FTC, dF/dx = .... what?

58. rld613

You are forsure ready to kill me!!!!!!!!!!!!!!!

59. rld613

You are forsure ready to kill me!!!!!!!!!!!!!!!

60. Hero

61. rld613

dF/dx= sinx^2/x

62. JamesJ

correct. Hence the derivative wrt (with respect to) x of the entire integral equals what?

63. pokemon23

hey james can you help me with math?

64. JamesJ

This thing ... what's it equal to?

65. rld613

sinx^2/x????? this is a guess

66. JamesJ

it shouldn't be guess. it is exactly right.

67. rld613

oh ya !!!!!!!!

68. rld613

That is what i thought in the first place but i left out by accident the squared

69. rld613

I can be so annoying sometimes

70. JamesJ

Now one more to make sure you really understand. Evaluate this: $\frac{d \ }{dx} \int_x^{x^2} \sqrt{t^3+1} \ dt$

71. rld613

oh u r so funny!!!

72. JamesJ

Now write out the steps. This problem isn't exactly the same as the others.

73. rld613

ok give me a sec

74. rld613

$d/dx=(\sqrt{x ^{6}+1}-\sqrt{x ^{3}+1})$

75. rld613

This one i guessed

76. rld613

I think it was suppossed to be dF/dx

77. JamesJ

The expression is equal to $\frac{d \ }{dx} \left( F(x^2) - F(x) \right) \ \ \ \ \ \ --(*)$ where by the Fund Theorem of Calculus, $dF/dx = \sqrt{x^3+1}$ Now given that, evaluate the first expression, (*).

78. JamesJ

hint: you need to use the chain rule.

79. rld613

ok so let me try that

80. rld613

james u gotta help me out on this one

81. JamesJ

you think about it for a bit and write the answer sometime in the next day when you know what's going on.

82. rld613

alrighty thanks james

83. rld613

I will call you back when i get the

84. rld613

85. rld613

$2x \sqrt{x ^{6}+1} - \sqrt{x ^{3}+1}$ Here is my answer. Thanks james for your help. I really appreciate it

86. JamesJ

that's it. Good

87. JamesJ

One more variation they might throw at you in the exam. Evaluate: $\frac{d^2 \ }{dx^2} \int_2^x \ln(\cos t) \ dt$

88. rld613

Oh this is very simple there is just one small catch that you have to differentiate it twice

89. rld613

d/dx(ln(cosx)) (1/cosx)*-sin(x)= -sin(x)/cos(x) -tan(x)

90. JamesJ

exactly, good.

91. rld613

I think that is the answer. I took my final exam already and it was basically simple except ofr a related rates problem. Going on vacation for a month So see u then. Thanks for your help james

92. rld613

oh u are here LOL

93. rld613

Thanks for helping me out

94. JamesJ

sure. have a good vacation.

95. rld613

U better take a vacation from OS tooo!!! :D

96. JamesJ

I might just do that.

97. rld613

Ya u need a break to refresh again

98. rld613

OS wldnt survive without James myinaya satellite and Hero