Evaluate the following improper integrals with infinite limits on the basis of their definition :
(a) `int_(2)^(oo)(xdx)/(sqrt((x^(2)-3)^(3))),`
(b) `int_(1)^(oo)(dx)/(x+x^(3)),`
(c ) `int_(0)^(oo)(xdx)/(sqrt((4x^2+1)^(3))),`
(d) `int_(1)^(oo)(dx)/(x^(2)(1+x)),`
(e) `int_(-oo)^(oo)(dx)/(x^(2)-6x+10),`
(f) `int_(0)^(oo)e^(-x)sinxdx`

Answer

(b) `(1)/(2)1n2;`
(c ) `1;`
(d) `1-1n2;` (e ) `pi;(f)(1)/(2)`.