Evaluate the following definite integral: ∫√(1 + cos x) dx, x ∈ [0, π/2 ]

Question

Evaluate the following definite integral:

\(\int\limits_a^{π/2}\sqrt{1+cos\,\text x}d\text x \)

Answer 1

\(\int\limits_0^{π/2}\sqrt{1+cos\,\text x}d\text x \) = \(\int\limits_0^{π/2}(\sqrt{1+cos\,\text x})\times\cfrac{\sqrt{1-cos\,\text x}}{\sqrt{1-cos\text x}}d\text x\)

Let 1 – cos x = t² hence sin x dx = 2 t dt

⇒ \(\int\limits_0^{π/2}(\sqrt{1+cos\,\text x})d\text x \)