I want to how to...

Questions
1

I want to how to solve the given integration

Screenshotfrom2019-06-2818-14-21.png
Screenshotfrom2019-06-2818-14-21.png1920Ă—1080 133 KB

2

Please post code that can be pasted into a notebook if you want to get a larger audience to help. Also choose a title that is informative.

3

\gamma = -(\alpha + 1/\alpha) ;
q = x^2 + (1 - x)^2 - x (1 - x) \gamma ;
p0 = \gamma / q ;
r = (1 + \alpha^2)/(1 - \alpha^2) ;

How to solve the following:

I = -1/r \int^{1}_{0} p0 * Log\[x\]*Log[1-x] dx

4

That is still not copyable Mathematica code! Please either indent 4 spaces so as to get a line of code; or select what you want to be formatted as copyable Mathematica code and click the first icon in the toolbar of the edit window here so as to indent the entire selected block.

For example, by doing that the first two lines of your input would look here like the following;

\[Gamma] = -(\[Alpha] + 1/\[Alpha]);
q = x^2 + (1 - x)^2 - x (1 - x) \[Gamma];
5

[Gamma] = -([Alpha] + 1/[Alpha]);
q = x^2 + (1 - x)^2 - x (1 - x)*[Gamma] ;
p0 = [Gamma]/q ;
r = (1 + [Alpha]^2)/(1 - [Alpha]^2) ;

Find the following

I6 = -1/r \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]\(p0*Log[x]*
    Log[1 - x] \(\(\[DifferentialD]\)\(x\)\(\ \)\)\)\)
6

$Version
(“12.0.0 for Microsoft Windows (64-bit) (April 6, 2019)”)

? = -(? + 1/?); 
q = x^2 + (1 - x)^2 - x (1 - x)*?; 
p0 = ?/q; 
r = (1 + ?^2)/(1 - ?^2);
func = -1/r*p0*Log[x]*Log[1 - x] // FullSimplify
sol=Integrate[func, {x, 0, 1}, Assumptions -> ? > 0]

(*for ? > 1 *)

(* (24*? - 12*EulerGamma*? - 2*Pi^2*? - 33*?^2 + 12*EulerGamma*?^2 - Pi^2*?^2 + 
  2*EulerGamma*Pi^2*?^2 + 9*?^3 - 15*?^2*Log[?] + 2*Pi^2*?^2*Log[?] - 
  18*?^2*Log[-1 + ?]*Log[?] + 12*EulerGamma*?^2*Log[-1 + ?]*Log[?] + 
  9*?^2*Log[?]^2 - 6*EulerGamma*?^2*Log[?]^2 + 12*EulerGamma*?^2*PolyLog[2, 1 - ?] + 
  18*?^2*PolyLog[2, ?^(-1)] - 12*EulerGamma*?^2*PolyLog[2, ?^(-1)] - 
  12*EulerGamma*?^2*PolyLog[2, (-1 + ?)/?] - 3*(-1 + ?)^2*?^2*
   Derivative[0, 0, 1, 0][Hypergeometric2F1][1, 2, 3, 1 - ?] + 
  12*(-1 + EulerGamma)*(-1 + ?)^2*Derivative[0, 0, 1, 0][Hypergeometric2F1Regularized][1, 
    2, 3, (-1 + ?)/?] + 6*?^2*Derivative[0, 0, 2, 0][Hypergeometric2F1][1, 2, 3, 
    1 - ?] - 12*?^3*Derivative[0, 0, 2, 0][Hypergeometric2F1][1, 2, 3, 1 - ?] + 
  6*?^4*Derivative[0, 0, 2, 0][Hypergeometric2F1][1, 2, 3, 1 - ?] - 
  12*Derivative[0, 0, 2, 0][Hypergeometric2F1Regularized][1, 2, 3, (-1 + ?)/?] + 
  24*?*Derivative[0, 0, 2, 0][Hypergeometric2F1Regularized][1, 2, 3, (-1 + ?)/?] - 
  12*?^2*Derivative[0, 0, 2, 0][Hypergeometric2F1Regularized][1, 2, 3, (-1 + ?)/?] + 
  6*?^2*Derivative[0, 1, 1, 0][Hypergeometric2F1][1, 2, 3, 1 - ?] - 
  12*?^3*Derivative[0, 1, 1, 0][Hypergeometric2F1][1, 2, 3, 1 - ?] + 
  6*?^4*Derivative[0, 1, 1, 0][Hypergeometric2F1][1, 2, 3, 1 - ?] - 
  12*Derivative[0, 1, 1, 0][Hypergeometric2F1Regularized][1, 2, 3, (-1 + ?)/?] + 
  24*?*Derivative[0, 1, 1, 0][Hypergeometric2F1Regularized][1, 2, 3, (-1 + ?)/?] - 
  12*?^2*Derivative[0, 1, 1, 0][Hypergeometric2F1Regularized][1, 2, 3, (-1 + ?)/?])/
 (12*?^2)*)

If we assume ?=2 then:

 sol[[1]]/. ? -> 2 // N
 (*  -0.486141 *)
 NIntegrate[func/.? -> 2, {x, 0, 1}]
 (* -0.486141 *)

Yours formula:

   4*PolyLog[3, 1 - ?] + 2*PolyLog[3, ?] - 
        4*PolyLog[2, 1 - ?]*Log[1 - ?] - 
        4*PolyLog[2, ?]*Log[1 - ?] - 
        4*Log[?]*Log[1 - ?]^2 + 
        Log[?]^2*Log[1 - ?] + 
        2*?^2*Log[1 - ?]/3 - 2 Zeta[3] /. ? -> 2 // N
  (*-0.486141 + 7.10543*10^-15 I*)

Simplify more:

   Log[1 - ?] Log[?]^2 + 4 PolyLog[3, 1 - ?] + 2 PolyLog[3, ?] - 2 Zeta[3] /. ? -> 2 // N
   (*-0.486141 + 2.66454*10^-15 I*)
7

I am also getting the same result but what one should get is the following which I am not getting.

Screenshot2019-06-28at9.19.28PM.png
Screenshot2019-06-28at9.19.28PM.png982Ă—126 7.34 KB

8

You STILL did not follow the instructions about formatting Mathematica code!

9

(1) How do you know they are different?

(2) Do you expect everyone to translate the image to Wolfram Language code? That’s not our task to perform.

10

The answer I am expecting is the following

I want an answer in this form

S2 = 4*PolyLog[3, 1 - \[Alpha]] + 2*PolyLog[3, \[Alpha]] - 
  4*PolyLog[2, 1 - \[Alpha]]*Log[1 - \[Alpha]] - 
  4*PolyLog[2, \[Alpha]]*Log[1 - \[Alpha]] - 
  4*Log[\[Alpha]]*(Log^2)[1 - \[Alpha]] + (Log^2)[\[Alpha]]*
   Log[1 - \[Alpha]] + 2*\[Pi]^2*Log[1 - \[Alpha]]/3 - 2 Zeta[3]

I am also not sure whether these two are the same or different but it’s highly likely that these two are not the same.

11

I would suggest checking numerically for several values of alpha.