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Challenge Micromass' big integral challenge

  1. Apr 20, 2016 #1

    micromass

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    Integrals are pretty interesting, and there are a lot of different methods to solve them. In this thread, I will give as a challenge 10 integrals. Here are the rules:

    • For a solution to count, the answer must not only be correct, but a detailed solution must also be given.
    • A correct answer would consist of the correct number (which may be infinite) or the statement that the given integral does not exist.
    • Any use of outside sources is allowed, but do not look up the integral directly. For example, it is ok to go check calculus textbooks for methods, but it is not allowed to type in the integral in wolframalpha.
    • If you previously encountered this integral and remember the solution, then you cannot participate with that specific integral.
    • All mathematical methods are allowed.

    Anyway, here we go:

    1. [itex]\int_0^1 \frac{\text{ln}(1+x)}{1+x^2}dx = \frac{\pi}{4}\text{ln}(\sqrt{2})[/itex] SOLVED BY Ssnow
    2. [itex]\int_0^\infty \frac{|\sin(x)|}{x}dx = +\infty[/itex] SOLVED BY PeroK
    3. [itex]\int_0^\infty \frac{\text{atan}(2016x) - \text{atan}(1916x)}{x}dx = \frac{\pi}{2}\text{ln} \left(\frac{2016}{1916}\right)[/itex] SOLVED BY Samy_A
    4. [itex]\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}}dx = \frac{\pi}{4}[/itex] SOLVED BY Ssnow
    5. [itex]\int_0^1 \sqrt{-\text{ln}(x)}dx = \frac{\sqrt{\pi}}{2}[/itex] SOLVED BY Ssnow
    6. [itex]\int_0^1 \frac{1-4x^5}{(x^5 - x + 1)^2}dx = 1[/itex] SOLVED BY Samy_A
    7. [itex]\int_0^{\pi/2} \text{acos}\left(\frac{\cos(x)}{1+2\cos(x)}\right)dx[/itex]
    8. [itex]\int_0^\infty \frac{\sin^{9}(x)}{x}dx = \frac{35}{256}\pi[/itex] SOLVED BY vela
    9. [itex]\int_0^\infty \frac{x^{1916}}{x^{2016} + 1}dx = \frac{1}{2016} \pi \frac{1}{\sin{\frac{1917 \pi}{2016}}}[/itex] SOLVED BY fresh_42
    10. [itex]\int_{\sqrt{2}}^\infty \frac{1}{x + x^{\sqrt{2}}}dx = -\ln \frac{\sqrt{2}}{{(1+{\sqrt 2}^{\sqrt{2}-1}})^{\sqrt{2}+1}}[/itex] SOLVED BY Samy_A
     
    Last edited: Apr 22, 2016
  2. jcsd
  3. Apr 20, 2016 #2

    Ssnow

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    I want propose the solution of 5. ##\int_{0}^{1}\sqrt{-\ln{x}}dx=\frac{\sqrt{\pi}}{2}##

    I started changing ##-\ln{x}=t^{2}## so ##x=e^{-t^{2}}## and ##dx=-2te^{-t^{2}}dt##, changing also the domain:

    ##\int_{0}^{\infty}2t^{2}e^{-t^{2}}dt##

    doing integration by parts ## t\rightarrow 1## and ## \int 2te^{-t^{2}}dt \rightarrow -e^{-t^{2}} ## so

    ##=-\lim_{z\rightarrow +\infty}te^{-t^{2}}|_{0}^{z}-\int_{0}^{\infty}-e^{-t^{2}}dt=0+\int_{0}^{\infty}e^{-t^{2}}dt=\frac{\sqrt{\pi}}{2}##

    The last is half the Gaussian integral. I declare I never see this integral before ... It is correct?
     
  4. Apr 20, 2016 #3

    micromass

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    Correct! Congratulations!
     
  5. Apr 20, 2016 #4

    PeroK

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    The integral in #2 must diverge. For non-negative integer ##n##:

    ##\int_{n\pi}^{(n+1)\pi} \frac{|sin x|}{x} dx > \int_{n\pi}^{(n+1)\pi} \frac{|sin x|}{(n+1)\pi} dx = \frac{2}{(n+1)\pi}##

    Hence:

    ##\int_{0}^{\infty} \frac{|sin x|}{x} dx > \sum_{n=0}^{\infty}\frac{2}{(n+1)\pi} = \frac{2}{\pi}\sum_{n=1}^{\infty} \frac{1}{n}##
     
  6. Apr 20, 2016 #5

    micromass

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    Correct!
     
  7. Apr 20, 2016 #6

    Samy_A

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    ##\displaystyle \int_0^\infty \frac{\text{atan}(2016x) - \text{atan}(1916x)}{x}dx##

    I use the general result ##\displaystyle \int_0^\infty \frac{f(ax)-f(bx)}{x} dx= (f(\infty) -f(0)) \log \frac{a}{b}## (proven using Leibnitz rule, see below).

    So ##\displaystyle \int_0^\infty \frac{\text{atan}(2016x) - \text{atan}(1916x)}{x}dx =(\text{atan}( \infty ) - \text{atan}( 0)) \log \frac{2016}{1916} = \frac{\pi}{2} \log \frac{2016}{1916}##.

    Now for ##\displaystyle \int_0^\infty \frac{f(ax)-f(bx)}{x} dx= (f(\infty) -f(0)) \log \frac{a}{b}## (*)
    Set ##\displaystyle I(y)= \int_0^\infty \frac{f(yx)-f(bx)}{x} dx##

    Notice that ##I(a)## is the integral we are trying to determine, and ##I(b)=0##.

    Then ##\displaystyle I'(y) = \int_0^\infty f'(yx) dx##. Let's do the substitution ##z=yx##, so that ##\displaystyle I'(y)= \frac{1}{y}\int_0^\infty f'(z) dz =\frac{1}{y} (f(\infty) -f(0))##. Let's name ##A=f(\infty) -f(0)##.
    We then have ##I'(y)=A \frac{1}{y}##, a differential equation with solution ##I(y)=A\log y + C##.
    Now, ##0=I(b)= A\log b + C##, so that ##C=-A\log b##.
    Finally, we have ##\displaystyle I(a)= A\log a + C = A\log a -A\log b= A \log \frac{a}{b}##, proving (*).

    (Remark: in this proof, I made some implicit assumptions about ##f## and ##a,b##, assumptions clearly satisfied by the explicit integral we had to solve here.)
     
  8. Apr 20, 2016 #7

    micromass

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    Very nice!
     
  9. Apr 20, 2016 #8
    Looks fun. I only just started integration, but I'm going to bookmark this for a future challenge!
     
  10. Apr 21, 2016 #9

    Ssnow

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    I propose the solution for the 4. ## \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin{x}}}{\sqrt{\sin{x}}+\sqrt{\cos{x}}}dx=\frac{\pi}{4}##

    The integral is ##\int_{0}^{\pi/2}\frac{1}{1+\sqrt{\cot{x}}}dx##, I start considering the substitution ##\cot{x}=t^{2}##, so the indefinite integral is:

    ## \int \frac{-2t}{(1+t)(1+t^{4})}dt##

    where ##\cot{x}=t^{2}## and ##dx=-\frac{1}{1+t^{4}}2tdt##. I started with partial fraction:

    ##\frac{-2t}{(1+t)(1+t^{4})}= \frac{A}{1+t}+\frac{Bt^{3}+Ct^{2}+Dt+E}{1+t^{4}} ##

    obtaining ##=\int \frac{1}{1+t}+\int \frac{-t^{3}+t^{2}-t-1}{1+t^{4}} ## that it is:

    ## \ln{(1+t)}-\frac{1}{4}\ln{(1+t^{4})} + \int \frac{t^{2}-t-1}{1+t^{4}}dt##

    for the last integral I repeat the partial fraction:

    ##\frac{t^{2}-t-1}{1+t^{4}}=\frac{At+B}{t^{2}+t\sqrt{2}+1}+\frac{Ct+D}{t^{2}-t\sqrt{2}+1}##

    and I find that ## A=-\frac{1}{\sqrt{2}},B=-(1+\sqrt{2})/2\sqrt{2}, C=1/\sqrt{2},D=(1-\sqrt{2})/2\sqrt{2}##, after I must adjust the two in order to have the derivative in the numerator, so obtaining the logaritm from both:

    ## -\frac{1}{2\sqrt{2}}\ln{(t^{2}+t\sqrt{2}+1)}## and from the second ##+\frac{1}{2\sqrt{2}}\ln{(t^{2}-t\sqrt{2}+1)}##,

    what remain are two integrals ##\int \frac{-1/(2\sqrt{2})}{t^{2}+t\sqrt{2}+1}dt## and ##\int \frac{1/(2\sqrt{2})}{t^{2}-t\sqrt{2}+1}dt##, they gives two factors with sum ##-\arctan{(1+\sqrt{2}t)}##

    so substituting ##t=\sqrt{\cot{x}}## and valutating between ##0## and ##\frac{\pi}{2}## (taking the limit where there are logaritm with ##\infty## because the ##\cot{0}=\infty##), I obtained the result ##\pi/2-\arctan{1}## that is ##\frac{\pi}{4}##. It is correct?
     
  11. Apr 21, 2016 #10

    micromass

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    That is the correct solution! Very well done!

    An easier (but much trickier!!) solution. Let ##I = \int_0^{\pi/2}\frac{\sqrt{\sin{x}}}{\sqrt{\sin{x}}+\sqrt{\cos{x}}}dx##. Perform substitution ##y = \pi/2 - x##, then
    [tex]I = \int_0^{\pi/2} \frac{\sqrt{\cos{y}}}{\sqrt{\sin{y}}+\sqrt{\cos{y}}}dy[/tex]
    Hence
    [tex]2I = \int_0^{\pi/2} \frac{\sqrt{\sin{x}}+\sqrt{\cos{x}}}{\sqrt{\sin{x}}+\sqrt{\cos{x}}}dx = \int_0^{\pi/2} dx = \frac{\pi}{2}[/tex]
    Thus, ##I = \frac{\pi}{4}##.
     
  12. Apr 21, 2016 #11

    Ssnow

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    I propose the solution of 1. ##\int_{0}^{1}\frac{\ln{(1+x)}}{1+x^{2}}=\frac{\pi}{4}\ln{\sqrt{2}}##

    I started substituting ##x=\tan{t}## so :

    ##\int_{0}^{\frac{\pi}{4}}\frac{\ln{(1+\tan{t})}}{1+\tan^{2}{t}}\frac{dt}{\cos^{2}{t}}##

    that is by the fundamental relation:

    ##=\int_{0}^{\frac{\pi}{4}}\ln{(1+\tan{t})}dt##

    Now ##1+\tan{t}=\frac{\cos{t}+\sin{t}}{\cos{t}}=\frac{sin{\left(\frac{\pi}{2}-t\right)}+\sin{t}}{\cos{t}}=\frac{2\sin{\pi/4}\cos{\left(t-\frac{\pi}{4}\right)}}{\cos{t}}=\frac{\sqrt{2}\cos{\left(t-\frac{\pi}{4}\right)}}{\cos{t}}##

    so

    ##=\int_{0}^{\frac{\pi}{4}}\ln{\sqrt{2}}dt+\int_{0}^{\frac{\pi}{4}}\ln{\cos{(t-\pi/4)}}dt-\int_{0}^{\pi/4}\ln{\cos{t}}dt##

    The second setting ##\frac{\pi}{4}-t=u## is

    ##=\int_{0}^{\frac{\pi}{4}}\ln{\cos{u}}du##

    so it cancel with the last and the integral is ##=\int_{0}^{\frac{\pi}{4}}\ln{\sqrt{2}}dt=\frac{\pi}{4}\ln{\sqrt{2}}##
     
  13. Apr 21, 2016 #12

    micromass

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    Correct again! Amazing that you came up with the substitution ##x = \tan{t}##. Very clever!!
     
  14. Apr 21, 2016 #13

    ShayanJ

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    So its not just me!

    Great thread micromass. Actually its starting to become an art exhibition!
     
  15. Apr 21, 2016 #14

    micromass

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    Half of my challenge problems are already solved. I did not expect this to happen so soon! The people who found these solutions are truly integral masters!

    Anyway, I wish to make a little adertisement now for the book containing these problems. I got these problems (except 6 which is from Apostol) from the beautiful book "Inside interesting integrals" from Paul J. Nahin. So if you want to learn tricky solutions or are up for a challenge. This is the book for you!!

    https://www.amazon.com/Inside-Interesting-Integrals-Substitutions-Undergraduate/dp/1493912763
     
  16. Apr 21, 2016 #15

    Samy_A

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    ##\displaystyle \int_{\sqrt{2}}^\infty \frac{1}{x + x^{\sqrt{2}}}dx##

    ##f(x)=\frac{1}{x + x^{\sqrt{2}}}##
    Doing a sort of partial fraction decomposition, ##\displaystyle f(x)=\frac{1}{x}- \frac{x^{\sqrt{2}-2}}{1+x^{\sqrt{2}-1}}##

    ##\displaystyle \int f(x) dx =\int \frac{1}{x} dx -\int \frac{x^{\sqrt{2}-2}}{1+x^{\sqrt{2}-1}}dx##
    ⇒ ##\displaystyle \int f(x) dx = \ln {x} - \frac{1}{\sqrt{2}-1} \int \frac{ d(x^{\sqrt{2}-1})}{1+x^{\sqrt{2}-1}} + C##
    ⇒ ##\displaystyle \int f(x) dx = \ln {x} - (\sqrt{2}+1)\ln(1+x^{\sqrt{2}-1}) + C##
    ⇒ ##\displaystyle \int f(x) dx =\ln \frac{x}{{(1+x^{\sqrt{2}-1}})^{\sqrt{2}+1}}+ C##

    ⇒ ##\displaystyle \int_{\sqrt{2}}^\infty f(x) dx = -\ln \frac{\sqrt{2}}{{(1+{\sqrt 2}^{\sqrt{2}-1}})^{\sqrt{2}+1}} \approx 1.50633 ##

    I used that ##\displaystyle \lim_{x \rightarrow +\infty} \ln \frac{x}{{(1+x^{\sqrt{2}-1}})^{\sqrt{2}+1}}=0## so that the term at ##+ \infty## vanishes.
    Proof for that limit:
    set ##y=x^{\sqrt{2}-1}##
    ⇒ ##y^{\sqrt{2}+1}=x^{(\sqrt{2}-1)(\sqrt{2}+1)}=x##
    ⇒ ##\displaystyle \lim_{x \rightarrow +\infty} \frac{x}{{(1+x^{\sqrt{2}-1}})^{\sqrt{2}+1}}= \lim_{y \rightarrow +\infty} \frac{y^{\sqrt{2}+1}}{(1+y)^{\sqrt{2}+1}} = \lim_{y \rightarrow +\infty} (\frac{y}{y+1})^{\sqrt{2}+1}=1##
    and hence the limit of the log of that nice expression will be 0.
     
    Last edited: Apr 21, 2016
  17. Apr 21, 2016 #16

    micromass

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    Very nice solution!
     
  18. Apr 21, 2016 #17

    vela

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    \begin{align*}
    \int_0^\infty \frac{\sin^9 x}{x}\,dx &= \frac 12 \int_{-\infty}^\infty \frac{\sin^9 x}{x}\,dx \\
    &= \frac 12 \frac{1}{(2i)^8} \int_{-\infty}^\infty \frac{1}{2i}\frac{(e^{ix}-e^{-ix})^9}{x}\,dx \\
    &= \frac {1}{512} \int_{-\infty}^\infty \frac{1}{2i}\left(\frac{e^{i9x}-9 e^{i7x} + 36 e^{i5x} - 84 e^{i3x} + 126 e^{ix} - 126 e^{-ix} + 84e^{-i3x} -36 e^{-i5x} + 9 e^{-i7x} - e^{-i9x}}{x}\right)\,dx \\
    &= \frac {1}{512} \int_{-\infty}^\infty \left(\frac{\sin 9x}{x} - 9 \frac{\sin 7x}{x} + 36 \frac{\sin 5x}{x} - 84 \frac{\sin 3x}{x} + 126\frac{\sin x}{x}\right)\,dx \\
    &= \frac{1}{512} (1-9+36-84+126)\int_{-\infty}^\infty \frac{\sin u}{u}\,du \\
    &= \frac{35}{256}\pi
    \end{align*}
     
  19. Apr 21, 2016 #18

    micromass

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    Very cool solution! Shows how complex numbers make life a lot easier!
     
  20. Apr 21, 2016 #19

    ShayanJ

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    Does that mean you had a more complicated method in mind?
    Although vela's method is indeed cool, I'd like to see other methods too.
     
  21. Apr 21, 2016 #20

    micromass

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    No, this is the method I had in mind. If you wish to find a solution without complex numbers, then you'll need to find another way to prove
    [tex]\sin^9x = \sin(9x) - 9\sin(7x) + 36\sin(5x) - 84\sin(3x) + 126\sin(x)[/tex]
    It is not very difficult to imagine how to prove or find this formula without complex numbers, you just use the addition formula on ##\sin(nx)## a few times. But using the complex exponentials seems to be a lot more elegant!
     
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