Evaluate the following Integral: ∫√(1 + cos x)/(1 - cos x)^3/2 dx, x ∈ [π/3, π/2 ]

Question

Evaluate the following Integral:

\(\int\limits_{\pi/3}^{\pi/2}\cfrac{\sqrt{1+cos\text x}}{(1-cos\text x)^{3/2}}d\text x\)

Answer 1

Let I = \(\int\limits_{\pi/3}^{\pi/2}\cfrac{\sqrt{1+cos\text x}}{(1-cos\text x)^{3/2}}d\text x\)

In the denominator, we can write

We also have,

When x = \(\cfrac{\pi}3\), t = cot\(\cfrac{(\frac{\pi}3)}{2}\) = cot \(\cfrac{\pi}6\) = √3

When x = \(\cfrac{\pi}2\), t = cot\(\cfrac{(\frac{\pi}2)}{2}\) = cot \(\cfrac{\pi}4\) = 1

So, the new limits are \(\sqrt3\) and 1.

Substituting this in the original integral,