Check the solution to "Evaluate the integrals 1−cos⁡x2dx"

ediculeN

Answered question

2021-10-06

Evaluate the integrals

\sqrt{\frac{1 - \cos x}{2}} d x

Answer & Explanation

Raheem Donnelly

Skilled2021-10-07Added 75 answers

Consider the given integral $\int_{0}^{2 π} \sqrt{\frac{1 - \cos x}{2}} d x$
Use the property of $\cos x = 1 - 2 \sin^{2} \frac{x}{2}$
$\int_{0}^{2 π} \sqrt{\frac{1 - \cos x}{2}} d x = \int_{0}^{2 π} \sqrt{\frac{1 - (1 - 2 \sin^{2} \frac{x}{2}} {2}}{d x}}$
$= \int_{0}^{2 π} \sqrt{\frac{2 \sin^{2} \frac{x}{2}}{2}} d x$
$= \int_{0}^{2 π} \sqrt{\frac{2 \sin^{2} \frac{x}{2}}{2}} d x$
$= \int_{0}^{2 π} \sqrt{\frac{2 \sin^{2} \frac{x}{2}}{2}} d x$
$= \int_{0}^{2 π} \sqrt{\sin^{2} \frac{x}{2}} d x$
As we know that $\sqrt{a^{2}} = | a |$
$\int_{0}^{2 π} \sqrt{\frac{1 - \cos x}{2}} d x = \int_{0}^{2 π} | \sin \frac{x}{2} | d x$
But $| \sin \frac{x}{2} | = \sin \frac{x}{2}$ because interval given $(0, 2 π)$
$\int_{0}^{2 π} \sqrt{\frac{1 - \cos x}{2}} d x = \int_{0}^{2 π} \sin \frac{x}{2} d x$
$= {[\frac{(- \cos \frac{x}{2})}{\frac{1}{2}}]}_{0}^{2 π}$
$= - 2 (\cos \frac{2 π}{2} = \cos 0)$
$= - 2 (\cos π - \cos 0)$

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