## monroe17 2 years 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

1. Spacelimbus

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

2. monroe17

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

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

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

5. monroe17

wait why and how did you rewrite it like that?

6. Spacelimbus

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

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

8. Spacelimbus

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

did it come from the 1/5x part?

10. Spacelimbus

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

11. monroe17

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

12. Spacelimbus

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

13. Spacelimbus

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

14. monroe17

oh lol.. okay

15. Spacelimbus

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

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

17. Spacelimbus

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

lol! okay :) give me a second

19. monroe17

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

20. Spacelimbus

perfect

21. Spacelimbus

You know ho to integrate that right?

22. monroe17

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

23. Spacelimbus

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

24. Spacelimbus

excuse my poor spelling (-; how to *

25. monroe17

isn't it just finding the antideriv?

26. Spacelimbus

exactly

27. Spacelimbus

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

28. Spacelimbus

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

29. monroe17

okay hold on :)

30. monroe17

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

31. Spacelimbus

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

32. Spacelimbus

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

33. monroe17

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

34. Spacelimbus

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

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

36. monroe17

oh i get it!

37. Spacelimbus

great (-: !

38. monroe17

Thank you, you're great at explaining :)

39. Spacelimbus

Appreciate it, you're welcome