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Math Help - Extremely simple but fun integral

  1. #1
    MHF Contributor Mathstud28's Avatar
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    Talking Extremely simple but fun integral

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

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

    Who is the fastest typer?
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    \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
    \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

    = \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
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by topsquark View Post
    If I said something like
    \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

    = \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...... \lim_{n\to\infty}\frac{admiration_1}{n}...Use it well!


    Edit:had to add that n_1 so you didnt get admiratio....
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    Lord of certain Rings
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    Quote Originally Posted by Mathstud28 View Post
    \int(\ln(x))^2dx

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

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

    \int t^2 e^t dt = e^t(t^2 - 2t + 2) = x((\ln x)^2 -2\ln x + 2)
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Isomorphism View Post
    With \ln x = t substitution,

    \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
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    \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.
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Jhevon View Post
    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!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    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
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  9. #9
    Math Engineering Student
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    Quote Originally Posted by Mathstud28 View Post
    \int(\ln(x))^2dx
    Sorry to say, but try to post interesting integrals. Your problems is easy, besides boring.
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  10. #10
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krizalid View Post
    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"
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  11. #11
    Math Engineering Student
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    Ohhh, I'm really sorry, I misread the thread title.
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  12. #12
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Krizalid View Post
    Ohhh, I'm really sorry, I misread the thread title.
    I...I...I was hurt...badly ...it's ok buddy..you are still my integrating hero =)
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  13. #13
    Math Engineering Student
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    I actually owe you an apologize, I'm sorry, really. (I had a computing test today, and it was terrible.)
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  14. #14
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Krizalid View Post
    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!
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  15. #15
    Lord of certain Rings
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    To Mathstud28,

    \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.
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