જો int sec ^2 x operatorname{cosec}^4 x , dx | vedclass

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(D) ધારો કે $I = \int \sec^2 x \operatorname{cosec}^4 x \, dx$. આપણે $\sec^2 x = \frac{1}{\cos^2 x}$ અને $\operatorname{cosec}^4 x = \frac{1}{\sin^4 x

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$1$

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(D) ધારો કે $I = \int \sec^2 x \operatorname{cosec}^4 x \, dx$. આપણે $\sec^2 x = \frac{1}{\cos^2 x}$ અને $\operatorname{cosec}^4 x = \frac{1}{\sin^4 x}$ લખી શકીએ. $I = \int \frac{1}{\cos^2 x \sin^4 x} \, dx = \int \frac{\sin^2 x + \cos^2 x}{\cos^2 x \sin^4 x} \, dx$. $I = \int \frac{\sin^2 x}{\cos^2 x \sin^4 x} \, dx + \int \frac{\cos^2 x}{\cos^2 x \sin^4 x} \, dx$. $I = \int \frac{1}{\cos^2 x \sin^2 x} \, dx + \int \operatorname{cosec}^4 x \, dx$. $I = \int \frac{\sin^2 x + \cos^2 x}{\cos^2 x \sin^2 x} \, dx + \int \operatorname{cosec}^2 x \operatorname{cosec}^2 x \, dx$. $I = \int (\sec^2 x + \operatorname{cosec}^2 x) \, dx + \int (1 + \cot^2 x) \operatorname{cosec}^2 x \, dx$. $I = 	an x - \cot x + \int \operatorname{cosec}^2 x \, dx + \int \cot^2 x \operatorname{cosec}^2 x \, dx$. ધારો કે $u = \cot x$,તો $du = -\operatorname{cosec}^2 x \, dx$. $I = 	an x - \cot x - \cot x - \int u^2 \, du$. $I = 	an x - 2 \cot x - \frac{u^3}{3} + C = 	an x - 2 \cot x - \frac{1}{3} \cot^3 x + C$. આપેલ સમીકરણ $-\frac{1}{3} \cot^3 x + k 	an x - 2 \cot x + C$ સાથે સરખાવતા,આપણને $k = 1$ મળે છે.

જો $\int \sec ^2 x \operatorname{cosec}^4 x \, dx = -\frac{1}{3} \cot ^3 x + k \tan x - 2 \cot x + C$ હોય,તો $k$ ની કિંમત શોધો.

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