>>543
まずxで積分する。∫[y,1] x*arctany dx = (1/2)(1-y^2)arctany
次にyで積分する。
∫[0,1] (1/2)(1-y^2)arctany dy
= (1/2)[(y-y^3/3)arctany][0,1] - (1/2)∫[0,1] {(y-y^3/3)/(1+y^2)}dy
= (1/2)(2/3)(π/4) - (1/2)∫[0,1] {y/(1+y^2)}dy + (1/6)∫[0,1] {y^3/(1+y^2)}dy
= π/12 - (1/2)∫[0,1] {y/(1+y^2)}dy + (1/6)∫[0,1] {y - y/(1+y^2)}dy
= π/12 - (2/3)∫[0,1] {y/(1+y^2)}dy + (1/6)∫[0,1] y dy
= π/12 - (1/3)[log(1+y^2)][0,1] + 1/12
= π/12 - (1/3)log2 + 1/12