Applying a suitable change of the variable, find the following definite integrals :
(a) `int_(0)^(2) (dx)/((sqrtx - 1) + sqrt((x + 1)^(3)))`
(b) `int_(0)^(a) (dx)/(x + sqrt(a^(2) - x^(2)))`
(c) `int_(1)^(2) (dx)/(x (1 + x^(4)))`
(d) `int_(sqrt((3a^(2)+ b^(2) )//2))^(sqrt((a^(2) + b^(2))//2)) (xdx)/(sqrt((x^(2) -a^(2))(b^(2) - x^(2))))`

Answer

(a) `(pi)/(6)`
(b) `(pi)/(4) `
(c) `(1)/(4) In "" (32)/(17)` (substitution `x^(4) = t`)
(d) `(pi)/(4) ` (substitution `x^(2) = a^(2) cos^(2) t + b^(2) sin^(2) t`)