# Evaluate the following Integral: ∫√((1 - x)/(1 + x))dx, x ∈ [0, 1]

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Evaluate the following Integral:

$\int\limits_0^1\sqrt{\cfrac{1-\text x}{1+\text x}}d\text x$

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Let I = $\int\limits_0^1\sqrt{\cfrac{1-\text x}{1+\text x}}d\text x$

As we have the trigonometric identity $\cfrac{1-cos2\theta}{1+cos2\theta}$ = tan2θ, to evaluate this integral we use x = cos 2θ

⇒ dx = –2sin(2θ)dθ (Differentiating both sides)

When x = 0, cos 2θ = 0 ⇒ 2θ =$\cfrac{\pi}2$ ⇒ θ =$\cfrac{\pi}4$

When x = 1, cos 2θ = 1 ⇒ 2θ = 0 ⇒ θ = 0

So, the new limits are $\cfrac{\pi}4$ and 0.

Substituting this in the original integral,