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(C) माना $I = \int \sin(101x) \sin^{99}x \, dx$. सर्वसमिका $\sin(101x) = \sin(100x + x) = \sin(100x)\cos x + \cos(100x)\sin x$ का उपयोग करने पर: $I =

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$1$

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(C) माना $I = \int \sin(101x) \sin^{99}x \, dx$. सर्वसमिका $\sin(101x) = \sin(100x + x) = \sin(100x)\cos x + \cos(100x)\sin x$ का उपयोग करने पर: $I = \int (\sin(100x)\cos x + \cos(100x)\sin x) \sin^{99}x \, dx$ $I = \int \sin(100x) \cos x \sin^{99}x \, dx + \int \cos(100x) \sin^{100}x \, dx$. प्रथम समाकलन के लिए खंडशः समाकलन (integration by parts) का उपयोग करने पर,$u = \sin(100x)$ और $dv = \cos x \sin^{99}x \, dx$ लें। अतः $du = 100 \cos(100x) \, dx$ और $v = \frac{\sin^{100}x}{100}$ प्राप्त होता है। $I = \sin(100x) \cdot \frac{\sin^{100}x}{100} - \int 100 \cos(100x) \cdot \frac{\sin^{100}x}{100} \, dx + \int \cos(100x) \sin^{100}x \, dx$. $I = \frac{\sin(100x) \sin^{100}x}{100} - \int \cos(100x) \sin^{100}x \, dx + \int \cos(100x) \sin^{100}x \, dx$. $I = \frac{\sin(100x) \sin^{100}x}{100} + C$. इसे दिए गए रूप $\frac{\sin(100x) \sin^{\lambda}x}{\mu} + C$ से तुलना करने पर,हमें $\lambda = 100$ और $\mu = 100$ प्राप्त होता है। अतः,$\frac{\lambda}{\mu} = \frac{100}{100} = 1$.

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