## monroe17 Group Title 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

1. Spacelimbus Group Title

This is a problem you want to solve by using the method of substitution. Got any idea how to do that?

2. monroe17 Group Title

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 ?

3. Spacelimbus Group Title

First I recommend you to rewrite the integral, like this: $\large \frac{1}{5}\int (\ln x)^5 \cdot \frac{1}{x}dx$

4. Spacelimbus Group Title

Yes, perfect @monroe17 !! You're pretty close from there.

5. monroe17 Group Title

wait why and how did you rewrite it like that?

6. Spacelimbus Group Title

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) $\large \frac{(\ln x)^5}{5x}= (\ln x)^5 \cdot \frac{1}{5x}$

7. monroe17 Group Title

question: (this may be a stupid question too) how'd you get 1/5 factored out?

8. Spacelimbus Group Title

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.

9. monroe17 Group Title

did it come from the 1/5x part?

10. Spacelimbus Group Title

exactly, I just thought the integral would look nicer without the unnecessary 5 in the denominator.

11. monroe17 Group Title

but then why did it stay 1/x ... why wouldn't it be x?

12. Spacelimbus Group Title

as an example:$\Large \int \frac{1}{5x}dx = \frac{1}{5} \int \frac{1}{x}dx$

13. Spacelimbus Group Title

Just a general law of multiplication $\large \frac{a}{b} \cdot \frac{c}{d}= \frac{ac}{bd}$

14. monroe17 Group Title

oh lol.. okay

15. Spacelimbus Group Title

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 (-:

16. monroe17 Group Title

oh okayy.. so now do i find the anti derivative?

17. Spacelimbus Group Title

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.

18. monroe17 Group Title

lol! okay :) give me a second

19. monroe17 Group Title

$\frac{ 1 }{ 5 }\int\limits_{?}^{?}(u)^5du$ this..?

20. Spacelimbus Group Title

perfect

21. Spacelimbus Group Title

You know ho to integrate that right?

22. monroe17 Group Title

uhm, i dont, this is where i get stuck the majority of the time. i dont get this part

23. Spacelimbus Group Title

would you know hot integrate this expression $\large \int x^3dx$ ?

24. Spacelimbus Group Title

excuse my poor spelling (-; how to *

25. monroe17 Group Title

isn't it just finding the antideriv?

26. Spacelimbus Group Title

exactly

27. Spacelimbus Group Title

with the general rule: $\Large \int x^ndx = \frac{1}{n+1}x^{n+1}+C$

28. Spacelimbus Group Title

You've managed to simplify your integral to exact that form, except that in your case it says $$u$$ rather than $$x$$

29. monroe17 Group Title

okay hold on :)

30. monroe17 Group Title

$\frac{ 1 }{ 5 }\int\limits_{?}^{?}\frac{ 1 }{ 6 }u^6+C$

31. Spacelimbus Group Title

exactly, now you can just multiply it out and you're done.

32. Spacelimbus Group Title

well, after backsubstitution you're done $u = \ln x$

33. monroe17 Group Title

what do you mean multiply it out? the 1/6 ?

34. Spacelimbus Group Title

by the way, I guess you mean the right thing, but you're spelling is a bit incorrect. You better write it like that: $\large \frac{1}{5}\int u^5du = \frac{1}{5}\left( \frac{1}{6}u^6+C\right)$

35. Spacelimbus Group Title

see the general rule of integration above again. the RHS doesn't include any integral sign anymore.

36. monroe17 Group Title

oh i get it!

37. Spacelimbus Group Title

great (-: !

38. monroe17 Group Title

Thank you, you're great at explaining :)

39. Spacelimbus Group Title

Appreciate it, you're welcome