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 one year ago
I need help with this step by step
Evaluate the indefinite integral.
\int \frac{(\ln(x))^4}{x} dx
by substitution
 one year ago
I need help with this step by step Evaluate the indefinite integral. \int \frac{(\ln(x))^4}{x} dx by substitution

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heradog
 one year ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{}^{}\frac{ \ln(x)^4 }{ x }dx\]

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2try \(u=ln(x), du =\frac{dx}{x}\) and you should get it in one step

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2visualize it as \[\int\frac{ \ln(x)^4 }{ x }dx=\int \ln^4(x)\frac{dx}{x}\]

heradog
 one year ago
Best ResponseYou've already chosen the best response.0pretend I'm a complete moron

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2are you familiar with "u  substitution"

SheldonEinstein
 one year ago
Best ResponseYou've already chosen the best response.0\(\ln x^4\) Why \(\ln ^4 x\) @satellite73

heradog
 one year ago
Best ResponseYou've already chosen the best response.0sorry, it's the whole thing to the 4th

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2it is the chain rule backwards

heradog
 one year ago
Best ResponseYou've already chosen the best response.0I understand that you pick one component to make u and that the derivative of that needs to be in the function it's what you do with it that I can't figure out

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2first off, lets get the answer and see why it works. what do you get when you take the derivative of \(\ln^3(x)\) ? by the chain rule you get \(3\ln(x)\times \frac{1}{x}=\frac{3\ln^2(x)}{x}\)

SheldonEinstein
 one year ago
Best ResponseYou've already chosen the best response.0Ok! \((\ln x)^4\) is what we have

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2this tells you \[\int\frac{3\ln^2(x)}{x}dx=\ln^3(x)\]

SheldonEinstein
 one year ago
Best ResponseYou've already chosen the best response.0I would not interupt and let @satellite73 answer.

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2so a usub is a way to undo the chain rule you have a composite function \(\ln^4(x)=(\ln(x))^4\) multiplied by the derivative of the "inside function" \(\frac{1}{x}\)

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2if you make the substitution \(u=\ln(x)\) then the derivative is \(\frac{1}{x}\) so the gimmick is to say \[u=\ln(x), du=\frac{1}{x}dx\]

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2then rewrite with only \(u\) terms and get \[\int u^4du\]

heradog
 one year ago
Best ResponseYou've already chosen the best response.0Ok I'm with you so far

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2now the anti derivative is relatively easy it is \(\frac{u^5}{5}\)

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2putting the \(u=\ln(x)\) back we get \[\frac{\ln^5(x)}{5}\]

heradog
 one year ago
Best ResponseYou've already chosen the best response.0why over 5? that's where I got lost

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2check that this works by differentiation, and you will see that this is correct

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2oh, over 5 because the derivative of \(u^5\) is \(5u^4\) but you want \(u^4\) and not \(5u^4\) so you have to divide by \(5\) to get it

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2\[\int x^n dx =\frac{x^{n+1}}{n+1}\]

heradog
 one year ago
Best ResponseYou've already chosen the best response.0I'm used to doing (1/5)u^5

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2check that the derivative of \[\frac{1}{5}\ln^5(x)\] is \[\frac{\ln^4(x)}{x}\] and you will see why the "u  substitution" works

SheldonEinstein
 one year ago
Best ResponseYou've already chosen the best response.0I did like this : \[\int \cfrac{ [\ln x]^4 dx }{x} \] Put \(\cfrac{1}{5} [\ln x]^5 = u \) \(\int du = u + c\) = \(\cfrac{1}{5} [\ln x]^5 + C\) \(\implies \cfrac{[\ln x]^4 dx}{x} = du\) Is this also correct @satellite73 ?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2it looks like you knew the answer at the start and then just said it look at this line \[\int du = u + c\] this says the integral is \(u\) which you knew at the beginning, probably by doing the u  sub in your head in other words, you knew what the answer was, but the u  sub is the way to get the answer

heradog
 one year ago
Best ResponseYou've already chosen the best response.0both of those really help! I'll try the next one and see what happens

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2the entire problem is to figure out the \[\frac{1}{5}\ln^5(x)\] part

SheldonEinstein
 one year ago
Best ResponseYou've already chosen the best response.0Ok! I got it. @heradog not me, @satellite73 was the one who helped. Best of luck for your journey and btw \(\mathbb{ I} \space \mathbb{LOVE} \space \mathbb{INTEGRALS} \)
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