Question

Evaluate the following definite integrals:

`int_0^1 sqrt((1 - x)/(1 + x)) "d"x`

Answer 1

Let I = `int_0^1 sqrt((1 - x)/(1 + x)) "d"x`    .......(1)

Put x = cos2θ   ........(2)

x	0	1
t	`pi/4`	0

DIfferentiate with respect to θ

dx = – 2sin²θdθ   ........(3)

Substitute (2) and (3) in (1), we get

(1) ⇒ I = `int_(pi/4)^0 sqrt((1 - cos 2theta)/(1 + cos 2theta)) (- 2 sin^2theta)"d"theta`

= `2int_0^4 (1 - cos 2theta)"d"theta`

= `2[theta - (sin 2theta)/2]_0^(pi/4)`

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

= `pi/2 - 1.

Consider:

`sqrt((1 - cos 2theta)/(1 + co 2theta)) (sin 2theta)`

= `sqrt((2sin^2theta)/(2cos^2theta)) (sin 2theta)`

= `sintheta/costheta (2sin thetacos theta)`

= 2sin²θ

= `2 ((1 - cos 2theta))/2`

= 1 – cos²θ