Evaluate the following definite integrals: _ 0 ^ 2 1+ dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\sqrt{1+\cos\text{x}}\text{ dx}$

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Answer

We use $1+\cos\text{x}=2\cos^2\frac{\text{x}}{2}$
$=\int_{0}^\limits{\frac{\pi}{2}}\sqrt{2\cos^2\frac{\text{x}}{2}}\text{ dx}$
$=\int_{0}^\limits{\frac{\pi}{2}}\sqrt{2}\cos\frac{\text{x}}{2}\text{ dx}$
$=\sqrt{2}\Big[2\sin\frac{\text{x}}{2}\Big]^{\frac{\pi}{2}}_0$
$=2\sqrt{2}\Big[\frac{1}{\sqrt{2}}\Big]$
$=2$

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