Question Number 101882 by bachamohamed last updated on 05/Jul/20

   ∫(1/((√x)+(√(x+1))+(√(x+2))))dx

Commented by M±th+et+s last updated on 05/Jul/20

and till now i am trying with this    (1/((√x)+(√(x+1))+(√(x+2))))=(1/(4x^3 −x^2 +6x−4))((x+1)(√(x+1))+(x−1)(√(x+2))−2(√x)(√(x+1))(√(x+2))+((1−x^2 )/(2(√x))))+(1/(2(√x)))  mybe i will find an elementary solution

Answered by Dwaipayan Shikari last updated on 05/Jul/20

∫(1/((√(4x+2))+(√(x+2))))dx                      {(√(x+1))+(√x)=(√(4x+2  )) x≠0   and x∈N  ∫(((√(4x+2))−(√(x+2)))/(3x))dx  ∫((√(4x+2))/(3x))−∫((√(x+2))/(3x))  (1/6)∫((t^2 dt)/((t^2 −2)/4))dt−(2/3)∫(p^2 /(p^2 −2))dp    {suppose (√(4x+2))=t  and (√(x+2))=p  (2/3)[∫(t^2 /(t^2 −2)) dt −∫(p^2 /(p^2 −2))dp]=(2/3)((t−(1/(2(√2)))log(((t−(√2))/(t+(√2)))))−(2/3)((p−(1/(2(√2)))log(((p−(√2))/(p+(√2)))))+C    =(2/3)((√(4x+2))−(1/(2(√2)))log((((√(4x+2))−(√2))/((√(4x+2))+(√2))))−(√(x+2))+(1/(2(√2)))log((((√(x+2))−(√2))/((√(x+2))+(√2)))))+Constant    But this is not a rigorous solution.  This only happens in this condition .    This is proved by Ramanujan  ((√x)+(√(x+1))=(√(4x+2))   x≠0  x∈N)

Commented byprakash jain last updated on 05/Jul/20

What is specific result from  Ramanujan that you referring to  google search on (√x)+(√(x+1))=(√(4x+2))  does not given anything.

Commented byDwaipayan Shikari last updated on 05/Jul/20

https://youtu.be/jGiK3NSsHro (watch this video of Michael Penn)

Commented byDwaipayan Shikari last updated on 05/Jul/20

sorry it will be x∈N

Commented byprakash jain last updated on 05/Jul/20

Ok. You misunderstood.  ⌊(√n)+(√(n+1))⌋=⌊(√(4n+2))⌋

Commented byDwaipayan Shikari last updated on 05/Jul/20

Yes sir. I am a high school student. Kindly forgive my misunderstanding