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    There must be some crazy series magic in that question.

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    (Original post by nahomyemane778)
    i dont understand this solution. why is it not just n!/ (n-6)!
    Because we have to factor in the fact that in what has been counted, there are 24 possibilities for each arrangement (due to rotation).
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    I'm glad the integral has caught the attention of several.

    (Original post by james22)
    Using a few approximations, I get it to be close to \dfrac{\sqrt{21+6e+17\pi+6\log 2}}{2\sqrt 6}

    Is this close at all? I know it isn't correct but it may be close (or not).
    That is not close to what I have - the end result is quite pleasing.
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    (Original post by Farhan.Hanif93)
    Solution 57

    Let the integral be I. Apply x\to -x to get  I=\displaystyle\int_{-1}^{1} \dfrac{1-x^4\tan x}{1+x^2} dx (by the oddity of tan), then add the two forms of I together and divide by 2 to obtain I = \displaystyle\int_{-1}^{1} \dfrac{1}{1+x^2} dx = \dfrac{\pi}{2}.
    Could someone please explain what is meant by x\to -x - iv seen it on many occasions with integrals on different threads but what does it actually mean (as x approaches -x or sometimes i see as x approaches x-pi/2 with trig functions in integrals) - and how has it been used to solve this integral?
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    (Original post by Lord of the Flies)
    I'm glad the integral has caught the attention of several.



    That is not close to what I have - the end result is quite pleasing.
    I suspect it might be zero by any chance?...
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    (Original post by nahomyemane778)
    Could someone please explain what is meant by x\to -x - iv seen it on many occasions with integrals on different threads but what does it actually mean (as x approaches -x or sometimes i see as x approaches x-pi/2 with trig functions in integrals) - and how has it been used to solve this integral?
    It's basically making the substitution u=-x then relableling u as x.
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    (Original post by nahomyemane778)
    Could someone please explain what is meant by x\to -x - iv seen it on many occasions with integrals on different threads but what does it actually mean (as x approaches -x or sometimes i see as x approaches x-pi/2 with trig functions in integrals) - and how has it been used to solve this integral?
    I'm pretty sure it should be written x \mapsto -x. It means making the substitution x=-u. It's notation which I, for some reason, really don't get on with.
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    (Original post by Hasufel)
    I suspect it might be zero by any chance?...
    In my increasing annoyance, I told Mathematica to evaluate the integral. It can't make heads or tails of the infinite one, but the integral from 0 to pi/2 is 0.476799233144858312460420290485 (I truncated just before Mathematica became unsure of the value).
    Series magic isn't the way forward, probably, because of the incredible oscillation.
    The next interval (pi/2 to pi) is 0.064451554294069389229981006180 02101588561.
    The integral from 0 to 52pi is 0.577148744550820213734020144632 0565614660,
    while the integral from 0 to 52pi-pi/2 is
    0.576830209101454331960082427303 8202792160.
    The integral from 0 to infinity is therefore probably a bit over 0.57.
    The function is even.
    Integrals from 0 to successive multiples of pi/2:
    Spoiler:
    Show
    {0, 0.476799233144858312460420290484 7781350219, \
    0.541250787438927701690401296664 7991509075, \
    0.542619332307639244035545342745 1088735121, \
    0.559349772494621474105780414969 9427811109, \
    0.556626603584655649292388983405 0088351455, \
    0.565493766122123397295311961516 9343890190, \
    0.562676095078534347464323940231 0069389261, \
    0.568578966612447199825629504506 6395219467, \
    0.566045227932802619356893383815 9638211929, \
    0.570433197969940212255808184154 6369943884, \
    0.568191590674342144709018734847 2936506823, \
    0.571670380349540425770877151993 1228113384, \
    0.569678411138494950767325854999 6731255104, \
    0.572554499738374251388374075776 4101777815, \
    0.570769130498798239568005707852 7597463855, \
    0.573217784519324126493243451567 9109640784, \
    0.571603399715959973051233247813 8893393236, \
    0.573733773731943616940415296102 7661614416, \
    0.572262135610363742518674110176 1912568195, \
    0.574146621618710532116460783210 4832898553, \
    0.572795457086043655722870689114 3069884352, \
    0.574484439852845447739890883169 0934372290, \
    0.573236063046481036377981851654 0372503327, \
    0.574765976033673308452133608038 8961371906, \
    0.573606195050152129995152610944 1547799898, \
    0.575004212608338428757862056830 8946453147, \
    0.573921507912050607008904188617 3220244197, \
    0.575208424577491125967865790131 5492027935, \
    0.574193339753013291721020561545 8448505811, \
    0.575385414654433165142430479592 8560286908, \
    0.574430103826177293030427498318 4207621527, \
    0.575540285499530832271293612161 8016357357, \
    0.574638174477325319505083129504 1348294511, \
    0.575676939534603575605439395119 4103885399, \
    0.574822469480126910094645889963 8202562481, \
    0.575798412224176031989879021635 0178500244, \
    0.574986843578767788652546186333 6387338162, \
    0.575907100150149290142212828790 9211912380, \
    0.575134360982105163829676099199 6510431023, \
    0.576004920687361774133089942718 9061029274, \
    0.575267488127953492509242586012 8938142457, \
    0.576093426071511752614064496031 7335555266, \
    0.575388232667583169272517450311 5517185292, \
    0.576173886366239614408638947907 9343591301, \
    0.575498245396981159640542052919 6703626071, \
    0.576247350792051064081074404979 0788053456, \
    0.575598896169047893016020380650 7862704314, \
    0.576314693726515960226696695116 9577039377, \
    0.575691331218929117207977585689 4427354170, \
    0.576376649666731507710667511116 9979939942, \
    0.575776517003615189027728619928 6375871445, \
    0.576433840125077861170803283715 2978755696, \
    0.575855274117342106720375886284 0522896990, \
    0.576486794549200363683939951686 0231330369, \
    0.575928303808473065229058071211 9315601647, \
    0.576535966759874931523619822179 7915465612, \
    0.575996208914720601460500249520 8194424719, \
    0.576581747988454281771702828020 7605256636, \
    0.576059510540954241104595874310 2674168219, \
    0.576624477307194834760234928499 1288725390, \
    0.576118661456556384941059149122 9830706214, \
    0.576664450041077843544206576832 0029308963, \
    0.576174056941212119626677029736 7334043355, \
    0.576701924602609246449106883923 6273892065, \
    0.576226043628627975535959040156 3310910284, \
    0.576737128084072754772625385424 5024488800, \
    0.576274926766470702908319619223 8966352456, \
    0.576770260863022222665563949897 2870742775, \
    0.576320976213835110609734918279 5843759324, \
    0.576801500418341936427835646261 2624528903, \
    0.576364431425151470879575291351 7756217264, \
    0.576831004510358281851024003530 2908868374, \
    0.576405505614893749706385286147 3772240888, \
    0.576858913845304653887142199468 0837350388, \
    0.576444389255990838253397534006 7199459745, \
    0.576885354319117661951524917980 3344869216, \
    0.576481253033074449918738060770 6930805485, \
    0.576910438916062115535592166959 9199116442, \
    0.576516250347166411236593695368 7628322107, \
    0.576934269322584326835950785093 5091223017, \
    0.576549519449327871967781872615 8044272854, \
    0.576956937305009138992766994645 8312697561, \
    0.576581185265850210194970049372 7872563241, \
    0.576978525890436867395707147383 9791131518, \
    0.576611360965788698926863547069 7360225187, \
    0.576999110382874904678423639266 5200338100, \
    0.576640149312298040922797007500 4121812158, \
    0.577018759240814728359420207240 3114498692, \
    0.576667643831776137695265323747 8804063786, \
    0.577037534837805735860794482410 0630215593, \
    0.576693929828843762774066263397 3476105495, \
    0.577055494123829544009370418456 5373288347, \
    0.576719085270366187714433584319 5181040609, \
    0.577072689202248205015448895082 5561334733, \
    0.576743181557814688195843633354 8653396512, \
    0.577089167834637720889817876272 3192936628, \
    0.576766284204082901645017828030 6253978732, \
    0.577104973883808351157888703093 4434477932, \
    0.576788453428268730547599796047 6176031561, \
    0.577120147703665075230456324153 3131937811, \
    0.576809744679792305738911330120 2575064226, \
    0.577134726483204264272022081737 0252445548, \
    0.576830209101454331960082427303 8202792160, \
    0.577148744550820213734020144632 0565614660}

    This is starting to get infuriating :P
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    (Original post by james22)
    It's basically making the substitution u=-x then relableling u as x.
    (Original post by bananarama2)
    I'm pretty sure it should be written x \mapsto -x. It means making the substitution x=-u. It's notation which I, for some reason, really don't get on with.
    Ok i still dont see how this works: sub in u=-x

     I=\displaystyle\int_{-1}^{1} \dfrac{u^4\tan u -1}{1+u^2} du

    Then if you relabel as x (why- i dont get why you can do this either- they are not the same)
    Then you add the integrals and divide by 2- this part i understand but my problem is why are you allowed to do everything before that?
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    (Original post by nahomyemane778)
    Ok i still dont see how this works: sub in u=-x

     I=\displaystyle\int_{-1}^{1} \dfrac{u^4\tan u -1}{1+u^2} du

    Then if you relabel as x (why- i dont get why you can do this either- they are not the same)
    Because the u is just a dummy variable - you'll get exactly the same result whether you integrate  I=\displaystyle\int_{-1}^{1} \dfrac{u^4\tan u -1}{1+u^2} du or  I=\displaystyle\int_{-1}^{1} \dfrac{x^4\tan x -1}{1+x^2} dx .
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    (Original post by nahomyemane778)
    Ok i still dont see how this works: sub in u=-x

     I=\displaystyle\int_{-1}^{1} \dfrac{u^4\tan u -1}{1+u^2} du

    Then if you relabel as x (why- i dont get why you can do this either- they are not the same)
    Then you add the integrals and divide by 2- this part i understand but my problem is why are you allowed to do everything before that?
    In indefinate integrals the variable you are integrating is irrelivent, the integral of x between 0 and 1 = the integral of u between 0 and 1 = the integral of between 0 and 1, provided you are integrating with respect to that variable.
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    (Original post by Smaug123)
    In my increasing annoyance, I told Mathematica to evaluate the integral. It can't make heads or tails of the infinite one, but the integral from 0 to pi/2 is 0.476799233144858312460420290485 (I truncated just before Mathematica became unsure of the value).
    Series magic isn't the way forward, probably, because of the incredible oscillation.
    The next interval (pi/2 to pi) is 0.064451554294069389229981006180 02101588561.
    The integral from 0 to 52pi is 0.577148744550820213734020144632 0565614660,
    while the integral from 0 to 52pi-pi/2 is
    0.576830209101454331960082427303 8202792160.
    The integral from 0 to infinity is therefore probably a bit over 0.57.
    The function is even.
    Integrals from 0 to successive multiples of pi/2:
    Spoiler:
    Show
    {0, 0.476799233144858312460420290484 7781350219, \
    0.541250787438927701690401296664 7991509075, \
    0.542619332307639244035545342745 1088735121, \
    0.559349772494621474105780414969 9427811109, \
    0.556626603584655649292388983405 0088351455, \
    0.565493766122123397295311961516 9343890190, \
    0.562676095078534347464323940231 0069389261, \
    0.568578966612447199825629504506 6395219467, \
    0.566045227932802619356893383815 9638211929, \
    0.570433197969940212255808184154 6369943884, \
    0.568191590674342144709018734847 2936506823, \
    0.571670380349540425770877151993 1228113384, \
    0.569678411138494950767325854999 6731255104, \
    0.572554499738374251388374075776 4101777815, \
    0.570769130498798239568005707852 7597463855, \
    0.573217784519324126493243451567 9109640784, \
    0.571603399715959973051233247813 8893393236, \
    0.573733773731943616940415296102 7661614416, \
    0.572262135610363742518674110176 1912568195, \
    0.574146621618710532116460783210 4832898553, \
    0.572795457086043655722870689114 3069884352, \
    0.574484439852845447739890883169 0934372290, \
    0.573236063046481036377981851654 0372503327, \
    0.574765976033673308452133608038 8961371906, \
    0.573606195050152129995152610944 1547799898, \
    0.575004212608338428757862056830 8946453147, \
    0.573921507912050607008904188617 3220244197, \
    0.575208424577491125967865790131 5492027935, \
    0.574193339753013291721020561545 8448505811, \
    0.575385414654433165142430479592 8560286908, \
    0.574430103826177293030427498318 4207621527, \
    0.575540285499530832271293612161 8016357357, \
    0.574638174477325319505083129504 1348294511, \
    0.575676939534603575605439395119 4103885399, \
    0.574822469480126910094645889963 8202562481, \
    0.575798412224176031989879021635 0178500244, \
    0.574986843578767788652546186333 6387338162, \
    0.575907100150149290142212828790 9211912380, \
    0.575134360982105163829676099199 6510431023, \
    0.576004920687361774133089942718 9061029274, \
    0.575267488127953492509242586012 8938142457, \
    0.576093426071511752614064496031 7335555266, \
    0.575388232667583169272517450311 5517185292, \
    0.576173886366239614408638947907 9343591301, \
    0.575498245396981159640542052919 6703626071, \
    0.576247350792051064081074404979 0788053456, \
    0.575598896169047893016020380650 7862704314, \
    0.576314693726515960226696695116 9577039377, \
    0.575691331218929117207977585689 4427354170, \
    0.576376649666731507710667511116 9979939942, \
    0.575776517003615189027728619928 6375871445, \
    0.576433840125077861170803283715 2978755696, \
    0.575855274117342106720375886284 0522896990, \
    0.576486794549200363683939951686 0231330369, \
    0.575928303808473065229058071211 9315601647, \
    0.576535966759874931523619822179 7915465612, \
    0.575996208914720601460500249520 8194424719, \
    0.576581747988454281771702828020 7605256636, \
    0.576059510540954241104595874310 2674168219, \
    0.576624477307194834760234928499 1288725390, \
    0.576118661456556384941059149122 9830706214, \
    0.576664450041077843544206576832 0029308963, \
    0.576174056941212119626677029736 7334043355, \
    0.576701924602609246449106883923 6273892065, \
    0.576226043628627975535959040156 3310910284, \
    0.576737128084072754772625385424 5024488800, \
    0.576274926766470702908319619223 8966352456, \
    0.576770260863022222665563949897 2870742775, \
    0.576320976213835110609734918279 5843759324, \
    0.576801500418341936427835646261 2624528903, \
    0.576364431425151470879575291351 7756217264, \
    0.576831004510358281851024003530 2908868374, \
    0.576405505614893749706385286147 3772240888, \
    0.576858913845304653887142199468 0837350388, \
    0.576444389255990838253397534006 7199459745, \
    0.576885354319117661951524917980 3344869216, \
    0.576481253033074449918738060770 6930805485, \
    0.576910438916062115535592166959 9199116442, \
    0.576516250347166411236593695368 7628322107, \
    0.576934269322584326835950785093 5091223017, \
    0.576549519449327871967781872615 8044272854, \
    0.576956937305009138992766994645 8312697561, \
    0.576581185265850210194970049372 7872563241, \
    0.576978525890436867395707147383 9791131518, \
    0.576611360965788698926863547069 7360225187, \
    0.576999110382874904678423639266 5200338100, \
    0.576640149312298040922797007500 4121812158, \
    0.577018759240814728359420207240 3114498692, \
    0.576667643831776137695265323747 8804063786, \
    0.577037534837805735860794482410 0630215593, \
    0.576693929828843762774066263397 3476105495, \
    0.577055494123829544009370418456 5373288347, \
    0.576719085270366187714433584319 5181040609, \
    0.577072689202248205015448895082 5561334733, \
    0.576743181557814688195843633354 8653396512, \
    0.577089167834637720889817876272 3192936628, \
    0.576766284204082901645017828030 6253978732, \
    0.577104973883808351157888703093 4434477932, \
    0.576788453428268730547599796047 6176031561, \
    0.577120147703665075230456324153 3131937811, \
    0.576809744679792305738911330120 2575064226, \
    0.577134726483204264272022081737 0252445548, \
    0.576830209101454331960082427303 8202792160, \
    0.577148744550820213734020144632 0565614660}

    This is starting to get infuriating :P
    Wolfram doesn't find anything interesting near those values, but could well be the case that the infinite tail of the integral is significant (i.e. the answer may actually be above 0.6).
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    (Original post by Lord of the Flies)
    I'm glad the integral has caught the attention of several.
    **** it, may as well jump on the bandwagon and add to the struggling masses!
    Spoiler:
    Show


    (Original post by Smaug123)
    0.577148744550820213734020144632 0565614660

    This is starting to get infuriating :P
    \gamma ?

    Problem 263 (Haven't actually done this but looks like fun -sorry)

    Let a, b, c, d be integers with a>b>c>d>0.

    Suppose that \displaystyle ac+bd=(b+d+a-c)(b+d-a+c).

    Prove that ab+cd is not prime.
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    (Original post by james22)
    In indefinate integrals the variable you are integrating is irrelivent, the integral of x between 0 and 1 = the integral of u between 0 and 1 = the integral of between 0 and 1, provided you are integrating with respect to that variable.
    k but when you change it integral is the wrong way around isnt it? there should be u^4tanu -1 not 1-u^4tanu as farhan did
    im very confused
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    (Original post by nahomyemane778)
    k but when you change it integral is the wrong way around isnt it? there should be u^4tanu -1 not 1-u^4tanu as farhan did
    im very confused
    Can you post the question and solution given? I cannot find them but I know his answer was correct (I think it was my question even).
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    (Original post by Jkn)
    Problem 263 (Haven't actually done this but looks like fun -sorry)

    Let a, b, c, d be integers with a>b>c>d>0.

    Suppose that \displaystyle ac+bd=(b+d+a-c)(b+d-a+c).

    Prove that ac+bd is not prime.
    Is the definiately typed correctly? I'm not great at number theory but seem to have got to an answer without much touble.
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    (Original post by james22)
    Is the definiately typed correctly? I'm not great at number theory but seem to have got to an answer without much touble.
    yeah i was a little confused too by the fact it was already factorised
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    (Original post by Jkn)
    **** it, may as well jump on the bandwagon and add to the struggling masses!
    Spoiler:
    Show


    \gamma ?

    Problem 263 (Haven't actually done this but looks like fun -sorry)

    Let a, b, c, d be integers with a>b>c>d>0.

    Suppose that \displaystyle ac+bd=(b+d+a-c)(b+d-a+c).

    Prove that ac+bd is not prime.
    Solution 263 (turns out the question was misstyped so this is a solution to a question knowone has asked, but I'll leave it up anyway)

    Assume, for a contradiction, that ac+bd is a prime. Using the inequality we can immediately deduce that b+d+a-c \neq 1

    Therefore b+d-a+c=1 (otherwise we have found 2 factors, contradicting the fact that ac+bd is prime).

    The equation then reduces to

    ac+bd=b+d+a-c

    which we can rearange then make use of the inequalities given to show that

    0=a(c-1)+b(d-1)+c-d) \geq 0+0+c-d=c-d>0

    which is a contradiction as required.
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    (Original post by james22)
    Is the definiately typed correctly? I'm not great at number theory but seem to have got to an answer without much touble.
    It certainly is! If you think it's right, go for it
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    (Original post by Jkn)
    It certainly is! If you think it's right, go for it
    Posted my solution above, is it valid?
 
 
 
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