int frac{operatorname{cosec}^2 x}{sec ^2 x} , | vedclass

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Question

(B) We have the integral $I = \int \frac{\operatorname{cosec}^2 x}{\sec ^2 x} \, dx$. Using the trigonometric identities $\operatorname{cosec}^2 x = \

Vedclass · Accepted Answer

$-\cot x - x$

Vedclass · Answer

(B) We have the integral $I = \int \frac{\operatorname{cosec}^2 x}{\sec ^2 x} \, dx$. Using the trigonometric identities $\operatorname{cosec}^2 x = \frac{1}{\sin^2 x}$ and $\sec^2 x = \frac{1}{\cos^2 x}$,we get: $I = \int \frac{\cos^2 x}{\sin^2 x} \, dx = \int \cot^2 x \, dx$. Using the identity $\cot^2 x = \operatorname{cosec}^2 x - 1$,we have: $I = \int (\operatorname{cosec}^2 x - 1) \, dx$. Integrating term by term,we get: $I = \int \operatorname{cosec}^2 x \, dx - \int 1 \, dx = -\cot x - x + C$. Thus,the correct option is $B$.

$\int \frac{\operatorname{cosec}^2 x}{\sec ^2 x} \, dx = $ . . . . . . $+ C$.

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