Hi, I found (redevelopped ?) some integral-identities for Pi and Log[2]. Starting from Eq. 34 of http://mathworld.wolfram.com/PiFormulas.html, i.e. Pi = 22/7 - Integrate[x^4(1 - x)^4/(1 + x^2), {x, 0, 1}], (in this special case 22/7 is a continued fraction of Pi) I found the more general relations: Pi = p_n/q_n -(-1)^n Integrate[x^(2n)(1 - x)^4/(1 + x^2), {x, 0, 1}], n>=0 where p_n/q_n is not a cont. fraction of Pi for n != 2. For increasing n the integral approaches quite slowly 0. The sequence p_n/q_n starts with 10/3, 47/15, 22/7, 1979/630, 10886/3465, 141511/45045, 141514/45045, 9622853/3063060, 45708802/14549535, 45708659/14549535, 150185878/47805615, 10512998863/3346393050, 3153902146/1003917915, 457315566013/145568097675, 14176787858138/4512611027925, 113414272443349/36100888223400, 14176786835558/4512611027925, 524541036307361/166966608033225, 40349314946642/12843585233325, 43012366079184317/13691261858724450, 924765931862392498/294362129962575675 Note the adjacent terms with the denominators 45045 and 14549535... -- For Log[2] I found the following (pls. note the strong similarity to the Pi integrals) Log[2] = 1 - Integrate[(1 - x)^2/(1 + x^2), {x, 0, 1}] Log[2] = 2/3 + Integrate[x^2(1 - x)^2/(1 + x^2), {x, 0, 1}] Log[2] = 7/10 - Integrate[x^4(1 - x)^2/(1 + x^2), {x, 0, 1}] Interestingly, 1, 2/3 and 7/10 are continued fractions of Log[2]. The general form is Log[2] = r_n/s_n -(-1)^n Integrate[x^(2n)(1 - x)^2/(1 + x^2), {x, 0, 1}], n>=0 but for n>2 r_n/s_n is not a continued fraction (at least I have not found any further). The sequence r_n/s_n starts with 1, 2/3, 7/10, 29/42, 25/36, 457/660, 541/780, 97/140, 9901/14280, 33181/47880, 1747/2520, 441871/637560, 96079/138600, 749293/1081080, 7244119/10450440, 1548577/2234232, 99917/144144, 8492321/12252240, 62847065/90666576, 32271119/46558512, 6615877847/9544494960 it looks that (2n+1)|s_n, and very often 6|s_n... And again, for increasing n the integral approaches quite slowly 0. I am quite sure that one could derive direct expressions for p_n,q_n,r_n,s_n through recurrence relations obtained by partial integration... -- In general Integrate[x^n(1 - x)^m/(1 + x^2)^p, {x, 0, 1}] , n,m,p>=0 gives a linear combination with rational coefficients of 1, Pi and Log[2]. The Pi and Log[2] come from the denominator in the integrals, so other polynomials (orthogonal ones?) in the numerator might give further interesting identities... Christoph