Evaluate the following Integral: ∫(cos x)/(cos x/2+sin x/2)^n dx, x ∈ [0, π/2]

Question

Evaluate the following Integral:

\(\int\limits_{0}^{\pi/2}\cfrac{cos\,\text x}{(cos\frac{\text x}2+ sin\frac{\text x}2)^n}d\text x \)

Answer 1

Let I = \(\int\limits_{0}^{\pi/2}\cfrac{cos\,\text x}{(cos\frac{\text x}2+ sin\frac{\text x}2)^n}d\text x \)

We can write,

Put cos\(\cfrac{\text x}2\) + sin\(\cfrac{\text x}2\) = t

⇒(-\(\cfrac12\) sin\(\cfrac{\text x}2\) +\(\cfrac12\)cos\(\cfrac{\text x}2\)) dx =dt

(Differentiating both sides)

⇒( cos\(\cfrac{\text x}2\) - sin\(\cfrac{\text x}2\) )dx = 2dt

When x = 0, t = cos 0 + sin 0 = 1

When x = \(\cfrac{\pi}2 \), t = cos\(\cfrac{(\frac{\pi}2)}2 \) + sin \(\cfrac{(\frac{\pi}2)}2 \)

= \(\cfrac{1}{\sqrt2}\) + \(\cfrac{1}{\sqrt2}\) = \(\sqrt2\)

So, the new limits are 1 and \(\sqrt2\).

Substituting this in the original integral,