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Π / 2 ∫ π / 6 C O S E C X Cot X 1 + C O S E C 2 X D X - Mathematics

Sum

\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]

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Solution

\[\int_\frac{\pi}{6}^\frac{\pi}{2} \frac{\ cosecx\ cotx}{1 + \ cosec^2 x} d x\]

\[ = \int_\frac{\pi}{6}^\frac{\pi}{2} \frac{\ cosx}{1 + \sin^2 x} d x\]

\[ = \left[ \tan^{- 1} \left(\ sinx \right) \right]_\frac{\pi}{6}^\frac{\pi}{2} \]

\[ = \tan^{- 1} 1 - \tan^{- 1} \frac{1}{2}\]

\[ = \tan^{- 1} \frac{1 - \frac{1}{2}}{1 + 1 \times \frac{1}{2}}\]

\[ = \tan^{- 1} \frac{1}{3}\]

Concept: Definite Integrals Problems
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 20 Definite Integrals
Revision Exercise | Q 59 | Page 123
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