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प्रश्न
Evaluate the following definite integrals:
`int_0^1 sqrt((1 - x)/(1 + x)) "d"x`
उत्तर
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
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