Print["Part (i)"]
m[y_] := Min[y, (Sqrt[2]) - y]
m[y_] := 
      If[0 ⩽ y ⩽ ((Sqrt[2])/2), 
        y, (Sqrt[2]) - y]
Plot[m[y], {y, 0, (Sqrt[2])}, 
        Ticks -> {{0, Sqrt[2]/2, Sqrt[2]}, {0, Sqrt[2]/2}}, 
        AspectRatio -> Automatic];
2  (1/2) (Sqrt[2]) (1/2) (Sqrt[2])

Print["Part (ii)"]
f[y1_, y2_] := (1/(2 - (y1^2) + (y2^2)))
Print["First integral"]
Integrate[f[y1, y2], y2, 
      Assumptions -> {0 < y1, y1 < (Sqrt[2]), 
          y2 > 0}]
Print["Second integral"]
%% /. y2 -> y1;
Integrate[%, y1]
Print["Third integral"]
%%%%% /. y2 -> (Sqrt[2]) - y1;
FullSimplify[Integrate[%, y1], y1 > 0]

    
FullSimplify[(Sqrt[((2 (Sqrt[ 2]))/(y1 + (Sqrt[2]))) - 1]) == (Sqrt[(((Sqrt[2]) - y1)/((Sqrt[2]) + y1))]), 0 < y1 < (Sqrt[2])]
Integrate[f[y1, y2], {y2, -y1, y1}, 
      Assumptions -> {0 < y1, y1 < (Sqrt[2]), 
          y2 > 0}]
Integrate[%, {y1, 0, ((Sqrt[2])/2)}]
Integrate[f[y1, y2], {y2, -m[y1], m[y1]}, 
      Assumptions -> {((Sqrt[2])/2) < y1 < (Sqrt[2])}]
Integrate[%, {y1, ((Sqrt[2])/2), (Sqrt[2])}]
ArcTan[((Sqrt[3])/3)]

Print["Part (iii)"]
Ψ[y1_, 
        y2_] := (1/(Sqrt[2])) {y1 - y2, y1 + y2}
h[{x1_, x2_}] := (1/(1 - x1 x2))
2  f[y1, y2] == 
      Simplify[h[Ψ[y1, y2]]]
Integrate[h[{x1, x2}], {x1, 0, 1}, {x2, 0, 1}]

Print["Part (vi)"]
Integrate[h[{x1, x2}], {x1, 0, 1}, Assumptions -> {0 < x2 < 1}] /. 
      x2 -> x
(∫_0^1) Log[1 - x]/x \[DifferentialD]x

Integrate[f[y1, y2], {y2, -((Sqrt[2]) - y1), (Sqrt[2]) - y1}, 
      Assumptions -> {0 < y1, y1 < (Sqrt[2]), 
          y2 > 0}]

      
Integrate[%, y1]
FullSimplify[Integrate[%%, y1]]
Integrate[%%%, {y1, (1/2) (Sqrt[2]), (Sqrt[2])}]

f[y1_, y2_] := (1/(2 - (y1^2) + (y2^2)))
∫f[y1, y2] \[DifferentialD]y2
FullSimplify[(1/2) (∂_y1) 
          ArcTan[(y1/(Sqrt[2 - (y1^2)]))]^2]
FullSimplify[-(∂_y1) 
          ArcTan[(((Sqrt[2]) - y1)/(Sqrt[2 - (y1^2)]))]^2]

sol = 
      Flatten[Solve[(y1/(Sqrt[2 - (y1^2)])) == v, 
          y1]]
y = 
      y1 /. sol[[2]]
FullSimplify[(Sqrt[2 - (y^2)]), v > 0]
FullSimplify[(∂_v) y]
% /%%

sol = 
      Flatten[Solve[(((Sqrt[2]) - y1)/(Sqrt[2 - (y1^2)])) == w, 
          y1]]
y = y1 /. sol
FullSimplify[(Sqrt[2 - (y^2)]), w > 0]
FullSimplify[(∂_w) y]
% /%%

TrigReduce[
      2 (((Cos[α]^2 - Sin[α]^2))^2)]
Simplify[
      2 (((Cos[α]^2 - Sin[α]^2))^2) == 
        2  Cos[2  α]^2]

Series[Log[1 - x]/x, {x, 0, 8}]