回答正确。

sorry, this:

u ^0.5 - u ^0.5

should be

tan (u) ^0.5 - tan (u) ^0.5

Set the answer = Y

Since tan (a-b) = (tan (a) – tan (b))/(1+tan (a) *tan (b)),

tan (pi/4-x) = (1-tan (x))/(1+tan (x))

set u = pi/4 – x, so, x=pi/4-u, tan (u) = (1-tan (x))/(1+tan (x)) , and dx = - du

tan (x) = (1-tan (pi/4-x))/(1+tan (pi/4+x)) = (1-tan (u)) / (1+ tan (u))

Y = integral [(1-tan (u)) / (1+ tan (u))] ^0.5/ ([(1-tan (u)) / (1+ tan (u))] ^0.5 + u ^0.5) (-du) (u is from pi/4 to 0)

= integral ([(1-tan (u)) / (1+ tan (u))] ^0.5 + u ^0.5 - u ^0.5)/ ([(1-tan (u)) / (1+ tan (u))] ^0.5 + u ^0.5) *du (u is from 0 to pi/4 now)

= intregral of 1 - y

2Y = pi/4 => Y = pi/8

pi/8