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integral of ((lnx)^5)/(5x)dx
please help step by step.. it's a practice problem for my exam this coming week.. and I need help reviewing because I've forgotten basically how to do these problems
 one year ago
 one year ago
integral of ((lnx)^5)/(5x)dx please help step by step.. it's a practice problem for my exam this coming week.. and I need help reviewing because I've forgotten basically how to do these problems
 one year ago
 one year ago

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SpacelimbusBest ResponseYou've already chosen the best response.2
This is a problem you want to solve by using the method of substitution. Got any idea how to do that?
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
u substitution..? I don't know how to do it.. it confuses me. the only thing i know is u= lnx and du= 1/x ?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
First I recommend you to rewrite the integral, like this: \[\large \frac{1}{5}\int (\ln x)^5 \cdot \frac{1}{x}dx \]
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
Yes, perfect @monroe17 !! You're pretty close from there.
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
wait why and how did you rewrite it like that?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
Well, first I did factor out the constants, then I just rewrote it to make the substitution more obvious. The rules are just algebraic. (Providing I understood your problem correctly) \[ \large \frac{(\ln x)^5}{5x}= (\ln x)^5 \cdot \frac{1}{5x} \]
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
question: (this may be a stupid question too) how'd you get 1/5 factored out?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
It's not too obvious if you're new to integrals, but it's a general rule of integration, you're integrating in the dimension / direction of \(dx\) therefore all numbers \( \in \mathbb{R} \) can be factored out. This will make your integral look 'easier' Or at least you wont be confused by the numbers anymore. Of course you must multiply them back in at the end. However, to some people this seems like uneasy extra work and they just skip that step, which is perfectly fine too.
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
did it come from the 1/5x part?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
exactly, I just thought the integral would look nicer without the unnecessary 5 in the denominator.
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
but then why did it stay 1/x ... why wouldn't it be x?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
as an example:\[\Large \int \frac{1}{5x}dx = \frac{1}{5} \int \frac{1}{x}dx \]
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
Just a general law of multiplication \[ \large \frac{a}{b} \cdot \frac{c}{d}= \frac{ac}{bd} \]
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
Most teachers provide that step, to some students  new to calculus  if you show them the integrals I posted above, they would say that they don't know how to integrate the first one, but the second one. It's more like to put empathy to the general structure of it, most of the time that's all what mathematics is about (:
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
oh okayy.. so now do i find the anti derivative?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
Use the substitution you have already figured out yourself, if you plug them into correctly you will see that it dramatically simplifies this devilish looking integral.
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
lol! okay :) give me a second
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
\[\frac{ 1 }{ 5 }\int\limits_{?}^{?}(u)^5du\] this..?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
You know ho to integrate that right?
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
uhm, i dont, this is where i get stuck the majority of the time. i dont get this part
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
would you know hot integrate this expression \[ \large \int x^3dx \] ?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
excuse my poor spelling (; how to *
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
isn't it just finding the antideriv?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
with the general rule: \[ \Large \int x^ndx = \frac{1}{n+1}x^{n+1}+C\]
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
You've managed to simplify your integral to exact that form, except that in your case it says \(u\) rather than \(x\)
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
\[\frac{ 1 }{ 5 }\int\limits_{?}^{?}\frac{ 1 }{ 6 }u^6+C\]
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
exactly, now you can just multiply it out and you're done.
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
well, after backsubstitution you're done \[ u = \ln x \]
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
what do you mean multiply it out? the 1/6 ?
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
by the way, I guess you mean the right thing, but you're spelling is a bit incorrect. You better write it like that: \[ \large \frac{1}{5}\int u^5du = \frac{1}{5}\left( \frac{1}{6}u^6+C\right) \]
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
see the general rule of integration above again. the RHS doesn't include any integral sign anymore.
 one year ago

monroe17Best ResponseYou've already chosen the best response.1
Thank you, you're great at explaining :)
 one year ago

SpacelimbusBest ResponseYou've already chosen the best response.2
Appreciate it, you're welcome
 one year ago
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