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

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(D) माना $I = \int \sec^2 x \operatorname{cosec}^4 x \, dx$. हम $\sec^2 x = \frac{1}{\cos^2 x}$ और $\operatorname{cosec}^4 x = \frac{1}{\sin^4 x}$ लिख

Vedclass · Accepted Answer

$1$

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(D) माना $I = \int \sec^2 x \operatorname{cosec}^4 x \, dx$. हम $\sec^2 x = \frac{1}{\cos^2 x}$ और $\operatorname{cosec}^4 x = \frac{1}{\sin^4 x}$ लिख सकते हैं। $I = \int \frac{1}{\cos^2 x \sin^4 x} \, dx = \int \frac{\sin^2 x + \cos^2 x}{\cos^2 x \sin^4 x} \, dx$. $I = \int \frac{\sin^2 x}{\cos^2 x \sin^4 x} \, dx + \int \frac{\cos^2 x}{\cos^2 x \sin^4 x} \, dx$. $I = \int \frac{1}{\cos^2 x \sin^2 x} \, dx + \int \operatorname{cosec}^4 x \, dx$. $I = \int \frac{\sin^2 x + \cos^2 x}{\cos^2 x \sin^2 x} \, dx + \int \operatorname{cosec}^2 x \operatorname{cosec}^2 x \, dx$. $I = \int (\sec^2 x + \operatorname{cosec}^2 x) \, dx + \int (1 + \cot^2 x) \operatorname{cosec}^2 x \, dx$. $I = 	an x - \cot x + \int \operatorname{cosec}^2 x \, dx + \int \cot^2 x \operatorname{cosec}^2 x \, dx$. माना $u = \cot x$,तो $du = -\operatorname{cosec}^2 x \, dx$. $I = 	an x - \cot x - \cot x - \int u^2 \, du$. $I = 	an x - 2 \cot x - \frac{u^3}{3} + C = 	an x - 2 \cot x - \frac{1}{3} \cot^3 x + C$. दिए गए समीकरण $-\frac{1}{3} \cot^3 x + k 	an x - 2 \cot x + C$ के साथ तुलना करने पर,हमें $k = 1$ प्राप्त होता है।

यदि $\int \sec ^2 x \operatorname{cosec}^4 x \, dx = -\frac{1}{3} \cot ^3 x + k \tan x - 2 \cot x + C$ है,तो $k$ का मान ज्ञात कीजिए।

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