Question

Evaluate the following : `int_1^oo 1/(sqrt(x)(1 + x))*dx`

Answer 1

Let I = `int_1^oo 1/(sqrt(x)(1 + x))*dx`

Put x = tan²t

∴ dx = `[2 tan t d/dt (tan t)]*dt`

= 2 tan t sec²t·dt

When x = `oo, tan^2t = oo    therefore t = pi/(2)`

When x = `1,  tan^2t = 1        therefore t = pi/(4)`

∴ I = `int_(pi/4)^(pi/2) (2tantsec^2t)/(sqrt(tan^2t) (1 + tan^2t))*dt`

= `int_(pi/4)^(pi/2) (2sec^2t)/(sec^2t)*dt`

= `2 int_(pi/4)^(pi/2) 1*dt = 2[t]_(pi/4)^(pi/2)`

= `2[pi/2 - pi/4]`

= `2[pi/4]`

= `pi/(2)`.