int frac{text{cosec}theta - cottheta}{text{co | vedclass

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Question

(A) We have the integral $I = \int \frac{	ext{cosec}	heta - \cot	heta}{	ext{cosec}	heta + \cot	heta} \, d	heta$. Multiply the numerator and den

Vedclass · Accepted Answer

$2	ext{cosec}	heta - 2\cot	heta - 	heta + c$

Vedclass · Answer

(A) We have the integral $I = \int \frac{	ext{cosec}	heta - \cot	heta}{	ext{cosec}	heta + \cot	heta} \, d	heta$. Multiply the numerator and denominator by $(	ext{cosec}	heta - \cot	heta)$: $I = \int \frac{(	ext{cosec}	heta - \cot	heta)^2}{	ext{cosec}^2	heta - \cot^2	heta} \, d	heta$. Since $	ext{cosec}^2	heta - \cot^2	heta = 1$,we get: $I = \int (	ext{cosec}	heta - \cot	heta)^2 \, d	heta = \int (	ext{cosec}^2	heta + \cot^2	heta - 2	ext{cosec}	heta \cot	heta) \, d	heta$. Using the identity $\cot^2	heta = 	ext{cosec}^2	heta - 1$: $I = \int (	ext{cosec}^2	heta + 	ext{cosec}^2	heta - 1 - 2	ext{cosec}	heta \cot	heta) \, d	heta = \int (2	ext{cosec}^2	heta - 1 - 2	ext{cosec}	heta \cot	heta) \, d	heta$. Integrating term by term: $I = 2(-\cot	heta) - 	heta - 2(-	ext{cosec}	heta) + c = 2	ext{cosec}	heta - 2\cot	heta - 	heta + c$.

$\int \frac{\text{cosec}\theta - \cot\theta}{\text{cosec}\theta + \cot\theta} \, d\theta = $

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