Question

\[\int\frac{ (lnx) ^{2} }{ x }dx\]

Answer 1

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

Answer 2

then

Answer 3

integral of 1/2*u*du

Answer 4

and

Answer 5

u srs?

Answer 6

or should I say u^2/2 srs?

Answer 7

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

Answer 8

i just don't know how to solve it

Answer 9

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

Answer 10

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!

Answer 11

yep

Answer 12

Were you replying to me or Algebraic?

Answer 13

you

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

@Callisto, it's the former.

Answer 19

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

Answer 20

nope:)

Answer 21

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

Answer 22

that clear it up @lambchamps ?

Answer 23

the second @Callisto

Answer 24

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

Answer 25

sorry it is the second guys

Answer 26

so it's
u = ln(x)
u^2 = (ln(x))^2
du = 1/x

go to town.

Answer 27

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

Answer 28

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

Answer 29

@Algebraic! please continue

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

dy = 1/x dx?

Answer 35

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

Answer 36

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

Answer 37

i mean (ln x)^2

Answer 38

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.

Answer 39

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

Answer 40

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

Answer 41

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

Answer 42

don't know please do tell

Answer 43

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

Answer 44

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

Answer 45

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.

Answer 46

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

Answer 47

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.

Answer 48

So what should u equal?

Answer 49

a?

Answer 50

Excellent!

Answer 51

and then

Answer 52

So now\[u =\ln x\]so then\[du =?\]

Answer 53

1/x

Answer 54

\[du =\frac{ 1 }{ x }dx\]OK?

Answer 55

ok sorry

Answer 56

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

Answer 57

@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!

Answer 58

@calculusfunctions effort appreciated.. thanks to both of you

Answer 59

sorry for interrupting .you may go on

Answer 60

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

Answer 61

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

Answer 62

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

Answer 63

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

Answer 64

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

Answer 65

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

Answer 66

So what does the integral look like after substitution?

Answer 67

dx would be xdu?

Answer 68

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?

Answer 69

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

Answer 70

@mikala, I'll help you right after I finish here.

Answer 71

because of the x.du

Answer 72

please tell me i'm right

Answer 73

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

Answer 74

Cancelled is a poor choice of words.

Answer 75

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

Answer 76

oh i'm sorry didn't see at first

Answer 77

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

Answer 78

yes

Answer 79

i see the 1/x dx

Answer 80

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

Answer 81

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

Answer 82

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

Answer 83

ok

Answer 84

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

Answer 85

\[\frac{ 1 }{ 3 }u ^{3} + c\]

Answer 86

Right!

Answer 87

Now what's the final step?

Answer 88

then substitute

Answer 89

ln x right?

Answer 90

Yes so can you please write the final answer now?

Answer 91

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

Answer 92

Perfect!!!

Answer 93

on more question is the c suppose to capitalized?

Answer 94

one*

Answer 95

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.

Answer 96

The most commonly used ones are c and k.