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Evaluate the following Integral:

\(\int\limits_{0}^{1/2}\cfrac{1}{(1+\text x^2)\sqrt{1-\text x^2}}d\text x \)

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Let I = \(\int\limits_{0}^{1/2}\cfrac{1}{(1+\text x^2)\sqrt{1-\text x^2}}d\text x \)

Put x = sin θ

⇒ dx = cos θ dθ (Differentiating both sides)

Also, \(\sqrt{1-\text x^2}=\sqrt{1-sin^2\theta}\) = cos \(\theta\)

When x = 0, sin θ = 0 ⇒ θ = 0

When x = \(\cfrac12\), sin θ = \(\cfrac12\) ⇒ θ = \(\cfrac{\pi}6\)

So, the new limits are 0 and \(\cfrac{\pi}6\)

Substituting this in the original integral,

Dividing numerator and denominator with cos2θ, we have

[∵ sec2θ = 1 + tan2θ]

Put tan θ = t

⇒ sec2θ dθ = dt (Differentiating both sides)

When θ = 0, t = tan 0 = 0

When θ = \(\cfrac{\pi}6\), t = tan \(\cfrac{\pi}6\) = \(\cfrac{1}{\sqrt3}\)

So, the new limits are 0 and \(\cfrac{1}{\sqrt3}\)

Substituting this in the original integral,

Recall

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