## anonymous 4 years ago $\int\frac{ (lnx) ^{2} }{ x }dx$

1. anonymous

u= 2ln(x) du = 2/x

2. anonymous

then

3. anonymous

integral of 1/2*u*du

4. anonymous

and

5. anonymous

u srs?

6. anonymous

or should I say u^2/2 srs?

7. anonymous

don't get it, because the answer is $\frac{ 1 }{3 }lnx ^{3}+c$

8. anonymous

i just don't know how to solve it

9. anonymous

don't think you can write it like that...

10. calculusfunctions

Alright, do you know the substitution rule for integrals? Also do you know the derivatives of logarithm functions? If you do, then we're off to a great start!

11. anonymous

yep

12. calculusfunctions

Were you replying to me or Algebraic?

13. anonymous

you

14. calculusfunctions

OK, so then when you see$\int\limits_{}^{}\frac{ \ln x ^{2} }{ x }dx$What do you see as the first potential step?

15. Callisto

Sorry to interrupt, is the question (i) $$\int \frac{lnx^2}{x}dx$$ or (ii) $$\int \frac{(lnx)^2}{x}dx$$? They are different...

16. calculusfunctions

Do you know your logarithm properties? for example$\ln x ^{n}=n \ln x$Do you know this property?

17. anonymous

yeah it's supposed to be (ln(x))^2

18. calculusfunctions

@Callisto, it's the former.

19. Callisto

For the first one, I don't think you can get (1/3)(lnx)^3 +C

20. anonymous

nope:)

21. anonymous

turn it to $\int\limits_{?}^{?}\frac{ 1 }{ x } \times \ln x ^{2} dx$

22. anonymous

that clear it up @lambchamps ?

23. anonymous

the second @Callisto

24. Callisto

You see... That's why...

25. anonymous

sorry it is the second guys

26. anonymous

so it's u = ln(x) u^2 = (ln(x))^2 du = 1/x go to town.

27. calculusfunctions

@Callisto, you are right but @lambchamps, Is the question written correctly before we proceed further?

28. anonymous

@calculusfunctions it's the second. based on what @Callisto have mentioned

29. anonymous

30. calculusfunctions

OK! so then we don't need the logarithm property I proposed earlier.

31. anonymous

that's it man, plug em in and integrate.

32. calculusfunctions

@lambchamps, do you now what the derivative of$y =\ln x$is?

33. anonymous

@calculusfunctions he or she is in calc. 2 so yeah probably. good question though.

34. anonymous

dy = 1/x dx?

35. calculusfunctions

Great, so then if$\int\limits_{}^{}\frac{ (\ln x)^{2} }{ x }dx$then in order to apply the substitution rule what should u equal?

36. anonymous

$=\int\limits_{}^{}\frac{ 1 }{ x } \times \ln x ^{2} dx$

37. anonymous

i mean (ln x)^2

38. calculusfunctions

First of all you keep confusing the issue by writing$\ln x ^{2}$instead of$(\ln x)^{2}$They are not the same!$(\ln x)^{2}\neq \ln x ^{2}$Do you understand? So we're not going to get anywhere until this gaffe is resolved.

39. calculusfunctions

OH, OK! Sorry, I see you did fix it.

40. anonymous

sorry it's $(\ln x)^{2}$

41. calculusfunctions

Do you notice that the derivative of ln x is in the integrand? So then what should u equal?

42. anonymous

43. anonymous

is it the $\frac{ (\ln x)^{n+1} }{ n+1 } ?$

44. calculusfunctions

Here are your options. Do you think u should equal a). ln x or b). 1/x c). (ln x)²

45. calculusfunctions

Which option do you think? a, b, or c? Just keep in mind that whichever option you choose, it's derivative must be in the integrand.

46. anonymous

the derivative of ln x is 1/x so i choose b

47. calculusfunctions

NO! I said that the derivative of the chose option must be in the integrand. NOT the antiderivative of the option must be in the integrand.

48. calculusfunctions

So what should u equal?

49. anonymous

a?

50. calculusfunctions

Excellent!

51. anonymous

and then

52. calculusfunctions

So now$u =\ln x$so then$du =?$

53. anonymous

1/x

54. calculusfunctions

$du =\frac{ 1 }{ x }dx$OK?

55. anonymous

ok sorry

56. calculusfunctions

So now if you substitute u and du into your integral, what do you have?

57. calculusfunctions

@integralsabiti, how is giving the solution helping the student who is trying to learn? I spent all this time trying to teach @lambchamps so that she can then do other similar problems with confidence, and you just came in wasted her time and my effort. NOT COOL!

58. anonymous

@calculusfunctions effort appreciated.. thanks to both of you

59. anonymous

sorry for interrupting .you may go on

60. calculusfunctions

@integralsabiti, thank you, no worries now that I know your intentions were genuine.

61. calculusfunctions

@lambchamps, are you still there? I'm still waiting for your response to my last question regarding your problem.

62. anonymous

so I would need to find the dx then substitute the value to it?

63. anonymous

ok calculausfunction after your done here would you help me on my problem pleases

64. calculusfunctions

So what do you you now have after substituting$u =\ln x$and$du =\frac{ 1 }{ x }dx$into$\int\limits_{}^{}\frac{ (\ln x)^{2} }{ x }dx$

65. calculusfunctions

Of course but let's hurry and finish this one first because I have to log out soon.

66. calculusfunctions

So what does the integral look like after substitution?

67. anonymous

dx would be xdu?

68. calculusfunctions

NO! If $u =\ln x$and$du =\frac{ 1 }{ x }dx$then$\int\limits\limits_{}^{}\frac{ (\ln x)^{2} }{ x }dx =\int\limits_{}^{}u ^{2}du$Do you see how?

69. anonymous

so the 1/x in (ln x)^2/x would be cancelled?

70. calculusfunctions

71. anonymous

because of the x.du

72. anonymous

73. calculusfunctions

No, the$\frac{ 1 }{ x }dx$is being replaced with du because$du =\frac{ 1 }{ x }dx$Please tell me you see that.

74. calculusfunctions

Cancelled is a poor choice of words.

75. anonymous

ok, but where did the 1/x go? the one below (ln x)^2

76. anonymous

oh i'm sorry didn't see at first

77. calculusfunctions

$\int\limits\limits_{}^{}\frac{ (\ln x)^{2} }{ x }dx$is exactly the same as$\int\limits_{}^{}(\frac{ 1 }{ x }∙(\ln x)^{2})dx$correct?

78. anonymous

yes

79. anonymous

i see the 1/x dx

80. anonymous

after finding that, then is the time to integrate, right?

81. calculusfunctions

So I replaced the factor of$\frac{ 1 }{ x }dx$with du because$du =\frac{ 1 }{ x }dx Yes, now you may integrate\[\int\limits_{}^{}u ^{2}du$

82. calculusfunctions

Sorry, I don't know what happened there let me try again.

83. anonymous

ok

84. calculusfunctions

Yes, now find the$\int\limits_{}^{}u ^{2}du$Can you do that please?

85. anonymous

$\frac{ 1 }{ 3 }u ^{3} + c$

86. calculusfunctions

Right!

87. calculusfunctions

Now what's the final step?

88. anonymous

then substitute

89. anonymous

ln x right?

90. calculusfunctions

91. anonymous

$\frac{ (\ln x)^{3} }{ 3 } + C$ ayt?

92. calculusfunctions

Perfect!!!

93. anonymous

on more question is the c suppose to capitalized?

94. anonymous

one*

95. calculusfunctions

That doesn't matter. C represents a constant. Whether you write c or C or k or K etc. is irrelevant. Just don't use x, y, or z.

96. calculusfunctions

The most commonly used ones are c and k.

97. anonymous

thank you very much Sheldon Cooper you're the best

98. calculusfunctions

HAHAHA! @lambchamps, thank you very much! "The Big Bang Theory" is my favourite show!

99. anonymous

i bet.. later dude

100. calculusfunctions

Later!