>>543
>Ｘ*arctanＹ
>これの二重積分です。
>範囲は ０≦Ｘ≦１　　　０≦Ｙ≦Ｘ

Ｙで積分　→　∫[0,x]arctany dy=∫[0,x](y)'arctany dy　←　部分積分して、
=(y*arctan)_[0,x] -∫[0,x] y/(1+y^2) dy　　←　(arctany)'=1/(1+y^2)を使う。
=x*arctanx -(1/2)log(1+x^2)　←　(log(1+y^2) )'=2y/(1+y^2) を使う

次にＸで積分　→　∫[0,1] (x*arctanx -(1/2)log(1+x^2) )dx

第一項を部分積分　→((1/2)x^2*arctanx)_[0,1] -∫[0,1](1/2)x^2/(1+x^2) dx
=(1/2)arctan(1) -(1/2)∫[0,1] (1-1/(1+x^2) )dx
=π/8 - (1/2)(1-arctan(1)) =π/4 -1/2　←∫ 1/(1+x^2) dx =arctanx を使う

第二項を積分　→(-1/2)∫log(1+x^2) =(-1/2) (x*log(1+x^2)_[0,1] -(-1/2)∫x(2x)/(1+x^2) dx
=(-1/2)log2 -(-1/2)2∫[0,1] x^2/(1+x^2) dx =(-1/2)log2 -(-1/2)2∫[0,1](1-1/(1+x^2) )dx
=(-1/2)log2 +(x-arctanx)_[0,1] =(-1/2)log2 +(1-π/4)

よって、π/4 -1/2 +(-1/2)log2 +1 -π/4 =(1/2)(1-log2)