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Solution for ∫ 3 π 2 π √ 1 − Cos 2 X D X - CBSE (Science) Class 12 - Mathematics

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Question

\[\int_\pi^\frac{3\pi}{2} \sqrt{1 - \cos2x}dx\]

Solution

\[\int_\pi^\frac{3\pi}{2} \sqrt{1 - \cos2x}dx\]
\[= \int_\pi^\frac{3\pi}{2} \sqrt{2 \sin^2 x}dx\]
\[ = \sqrt{2} \int_\pi^\frac{3\pi}{2} \left| \sin x \right|dx\]
\[ = - \sqrt{2} \int_\pi^\frac{3\pi}{2} \sin x\ dx \left( \sin x < 0 for\ \pi \leq x \leq 2\pi \right)\]

\[= - \sqrt{2}\left( - \cosx \right) |_\pi^\frac{3\pi}{2} \]
\[ = \sqrt{2}\left( \cos\frac{3\pi}{2} - cos\pi \right)\]
\[ = \sqrt{2} \left[ 0 - \left( - 1 \right) \right]\]
\[ = \sqrt{2} \times 1\]
\[ = \sqrt{2}\]

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Solution ∫ 3 π 2 π √ 1 − Cos 2 X D X Concept: Properties of Definite Integrals.
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