# Thread: Extremely simple but fun integral

1. ## Extremely simple but fun integral

$\displaystyle \int(\ln(x))^2dx$

First one to get this really easy integral wins....uhm.....wins...my admiration!

Who is the fastest typer?

2. Originally Posted by Mathstud28
$\displaystyle \int(\ln(x))^2dx$

First one to get this really easy integral wins....uhm.....wins...my admiration!

Who is the fastest typer?
If I said something like
$\displaystyle \int \left ( ln(x) \right ) ^2~dx = \int \left ( (x - 1)^2 - (x - 1)^3 + \frac{11(x - 1)^4)}{12} - \frac{5(x - 1)^5}{6} + ~...~ \right ) ~dx$

$\displaystyle = \frac{15}{4}x - \frac{77}{12}x^2 + \frac{107}{18}x^3 - \frac{13}{4}x^4 + \frac{61}{60}x^5 - \frac{5}{36}x^6 + ~...$
you'd probably get mad at me, wouldn't you?

-Dan

3. Originally Posted by topsquark
If I said something like
$\displaystyle \int \left ( ln(x) \right ) ^2~dx = \int \left ( (x - 1)^2 - (x - 1)^3 + \frac{11(x - 1)^4)}{12} - \frac{5(x - 1)^5}{6} + ~...~ \right ) ~dx$

$\displaystyle = \frac{15}{4}x - \frac{77}{12}x^2 + \frac{107}{18}x^3 - \frac{13}{4}x^4 + \frac{61}{60}x^5 - \frac{5}{36}x^6 + ~...$
you'd probably get mad at me, wouldn't you?

-Dan
Haha..congratulations you gave an answer...but not being the easy answer you win the secondary portion of my admiration......$\displaystyle \lim_{n\to\infty}\frac{admiration_1}{n}$...Use it well!

4. Originally Posted by Mathstud28
$\displaystyle \int(\ln(x))^2dx$

First one to get this really easy integral wins....uhm.....wins...my admiration!

Who is the fastest typer?
With $\displaystyle \ln x = t$ substitution,

$\displaystyle \int t^2 e^t dt = e^t(t^2 - 2t + 2) = x((\ln x)^2 -2\ln x + 2)$

5. Originally Posted by Isomorphism
With $\displaystyle \ln x = t$ substitution,

$\displaystyle \int t^2 e^t dt = e^t(t^2 - 2t + 2) = x((\ln x)^2 -2\ln x + 2)$
Hooray! You win this...♥...my heart...in a completely platonic math sort of way

6. Originally Posted by Mathstud28
$\displaystyle \int(\ln(x))^2dx$

First one to get this really easy integral wins....uhm.....wins...my admiration!

Who is the fastest typer?
integration by parts. u = (ln(x))^2, dv/dx = 1. this kills it pretty quickly.

7. Originally Posted by Jhevon
integration by parts. u = (ln(x))^2, dv/dx = 1. this kills it pretty quickly.
Aww...Jhevon...two minutes too late...but...as a prize for giving you thoughts you get this ...that is my used heart...a little worn by atherosclerosis..but still usable in times of sudden heart failure!

8. Originally Posted by Mathstud28
Aww...Jhevon...two minutes too late...but...as a prize for giving you thoughts you get this ...that is my used heart...a little worn by atherosclerosis..but still usable in times of sudden heart failure!
well, you asked the question while i was away from my pc. so it's not my fault.

luckily my heart is in great shape. so chances are, i won't need to use yours, hehe

if i may dare say, i believe my method is the fastest so far. so i don't feel too bad

9. Originally Posted by Mathstud28
$\displaystyle \int(\ln(x))^2dx$
Sorry to say, but try to post interesting integrals. Your problems is easy, besides boring.

10. Originally Posted by Krizalid
Sorry to say, but try to post interesting integrals. Your problems is easy, besides boring.
In mathstud's defense, he did say it was "extremely easy"

12. Originally Posted by Krizalid
I...I...I was hurt...badly ...it's ok buddy..you are still my integrating hero =)

13. I actually owe you an apologize, I'm sorry, really. (I had a computing test today, and it was terrible.)

14. Originally Posted by Krizalid
I actually owe you an apologize, I'm sorry really. (I had a computing test today, and it was terrible.)
No really don't worry about it...it was just a remark...you are the one that made me start taking integration seriously!

So really thank you!

15. To Mathstud28,

$\displaystyle \int\limits_0^{\frac {\pi }{4}} {\ln \left( {1 + \tan x} \right)dx}$

It appeared quite a few times in a few old entrance exams in the country.