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Question: Evaluate : \(\int_{–\pi}^{199\pi}\sqrt{\left( \frac{1 - \cos 2x}{2} \right)}\)dx –...

Evaluate : ∫–π199π(1−cos⁡2x2)\int_{–\pi}^{199\pi}\sqrt{\left( \frac{1 - \cos 2x}{2} \right)}dx –

A

200

B

400

C

–200

D

–400

Answer

400

Explanation

Solution

Let I = ∫–π199π(1−cos⁡2x2)\int_{–\pi}^{199\pi}\sqrt{\left( \frac{1 - \cos 2x}{2} \right)}dx

= ∫−π199π∣sin⁡x∣\int_{- \pi}^{199\pi}{|\sin x|}dx

= (199 – (–1)) ∫0π∣sin⁡x∣\int_{0}^{\pi}{|\sin x|} dx

(Q | sin x | is periodic with period p)

= 200 ∫0πsin⁡x\int_{0}^{\pi}{\sin x}dx

= 200 (– cos x)

= 200 (1 – (–1)) = 400.