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 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
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

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1u substitution..? I don't know how to do it.. it confuses me. the only thing i know is u= lnx and du= 1/x ?

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

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1wait why and how did you rewrite it like that?

Spacelimbus
 one year ago
Best ResponseYou've already chosen the best response.2Well, 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) \[ \large \frac{(\ln x)^5}{5x}= (\ln x)^5 \cdot \frac{1}{5x} \]

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1question: (this may be a stupid question too) how'd you get 1/5 factored out?

Spacelimbus
 one year ago
Best ResponseYou've already chosen the best response.2It'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.

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1did it come from the 1/5x part?

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1but then why did it stay 1/x ... why wouldn't it be x?

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

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

Spacelimbus
 one year ago
Best ResponseYou've already chosen the best response.2Most 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 (:

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1oh okayy.. so now do i find the anti derivative?

Spacelimbus
 one year ago
Best ResponseYou've already chosen the best response.2Use 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.

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1lol! okay :) give me a second

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1\[\frac{ 1 }{ 5 }\int\limits_{?}^{?}(u)^5du\] this..?

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1uhm, i dont, this is where i get stuck the majority of the time. i dont get this part

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

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1isn't it just finding the antideriv?

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

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1\[\frac{ 1 }{ 5 }\int\limits_{?}^{?}\frac{ 1 }{ 6 }u^6+C\]

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

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1what do you mean multiply it out? the 1/6 ?

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

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

monroe17
 one year ago
Best ResponseYou've already chosen the best response.1Thank you, you're great at explaining :)

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