int e^{x operatorname{cosec} x} cdot operator | vedclass

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Question

(B) Let $I = \int e^{x \operatorname{cosec} x} \cdot \operatorname{cosec} x \cdot (1 - x \cot x) \, dx$. Substitute $t = x \operatorname{cosec} x$. Di

Vedclass · Accepted Answer

$e^{x \operatorname{cosec} x} + c$

Vedclass · Answer

(B) Let $I = \int e^{x \operatorname{cosec} x} \cdot \operatorname{cosec} x \cdot (1 - x \cot x) \, dx$. Substitute $t = x \operatorname{cosec} x$. Differentiating with respect to $x$ using the product rule: $\frac{dt}{dx} = x \cdot \frac{d}{dx}(\operatorname{cosec} x) + \operatorname{cosec} x \cdot \frac{d}{dx}(x)$ $\frac{dt}{dx} = x(-\operatorname{cosec} x \cot x) + \operatorname{cosec} x(1)$ $\frac{dt}{dx} = \operatorname{cosec} x(1 - x \cot x)$ Therefore,$dt = \operatorname{cosec} x(1 - x \cot x) \, dx$. Substituting these into the integral: $I = \int e^t \, dt = e^t + c$ Substituting back $t = x \operatorname{cosec} x$: $I = e^{x \operatorname{cosec} x} + c$. Thus,option $B$ is correct.

$\int e^{x \operatorname{cosec} x} \cdot \operatorname{cosec} x \cdot(1-x \cot x) \, dx =$

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