int operatorname{cosec}^5 x , dx = | vedclass

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(C) माना $I = \int \operatorname{cosec}^5 x \, dx = \int \operatorname{cosec}^3 x \cdot \operatorname{cosec}^2 x \, dx$ है। खंडशः समाकलन (integration

Vedclass · Accepted Answer

$-\frac{\operatorname{cosec}^3 x \cot x}{4} - \frac{3}{8} \operatorname{cosec} x \cot x + \frac{3}{8} \log \left|	an \frac{x}{2}ight| + c$

Vedclass · Answer

(C) माना $I = \int \operatorname{cosec}^5 x \, dx = \int \operatorname{cosec}^3 x \cdot \operatorname{cosec}^2 x \, dx$ है। खंडशः समाकलन (integration by parts) का उपयोग करते हुए, $u = \operatorname{cosec}^3 x$ और $dv = \operatorname{cosec}^2 x \, dx$ लें। तब $du = 3 \operatorname{cosec}^2 x (-\operatorname{cosec} x \cot x) \, dx$ और $v = -\cot x$ होगा। $I = \operatorname{cosec}^3 x(-\cot x) - \int (-\cot x) (-3 \operatorname{cosec}^3 x \cot x) \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3 \int \operatorname{cosec}^3 x \cot^2 x \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3 \int \operatorname{cosec}^3 x (\operatorname{cosec}^2 x - 1) \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3 \int \operatorname{cosec}^5 x \, dx + 3 \int \operatorname{cosec}^3 x \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3I + 3I_1$, जहाँ $I_1 = \int \operatorname{cosec}^3 x \, dx$ है। $4I = -\operatorname{cosec}^3 x \cot x + 3I_1$। $I_1 = \int \operatorname{cosec}^3 x \, dx$ के लिए, खंडशः समाकलन का उपयोग करते हुए: $I_1 = -\operatorname{cosec} x \cot x - \int \operatorname{cosec} x(\operatorname{cosec}^2 x - 1) \, dx = -\operatorname{cosec} x \cot x - I_1 + \log|	an \frac{x}{2}|$ $2I_1 = -\operatorname{cosec} x \cot x + \log|	an \frac{x}{2}|$। $I_1$ का मान $4I$ के समीकरण में रखने पर: $4I = -\operatorname{cosec}^3 x \cot x + 3 \left( -\frac{1}{2} \operatorname{cosec} x \cot x + \frac{1}{2} \log|	an \frac{x}{2}| ight)$ $I = -\frac{\operatorname{cosec}^3 x \cot x}{4} - \frac{3}{8} \operatorname{cosec} x \cot x + \frac{3}{8} \log|	an \frac{x}{2}| + C$।

$\int \operatorname{cosec}^5 x \, dx =$

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