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If \(I = \int\limits^{\frac\pi2}_{\frac{-\pi}2} (4\lambda sin^4x + 54 sin^5x + 72 sin^7x)dx\) , then I is proportional tothe answers are a and c |
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Answer» Correct option is (D) all of these \(I = \int\limits^{\frac\pi2}_{\frac{-\pi}2} (4\lambda sin^4x + 54 sin^5x + 72 sin^7x)dx\) \(= 2 \int\limits^\frac\pi2_0 4\lambda sin^4 x \,dx\) \( = 8\lambda \frac{\Gamma\left(\frac52\right) + \Gamma\left(\frac12\right)}{2\Gamma(3)}\) (By gamma beta function) \(= 8\lambda \frac{\frac32.\frac12\,\Gamma\left(\frac12\right) \Gamma\left(\frac12\right)}{2\times 2!}\) \((\because \Gamma (n + 1) = n \Gamma (n))\) \(= \frac{3\lambda}{2}\sqrt{\pi}. \sqrt\pi\) \((\because \Gamma\left(\frac12\right) = \sqrt\pi )\) \(= \frac{3\pi\lambda}{2}\) ∴ \(I \propto \lambda\) (∵ \(\frac{3\pi}{2}\) is constant) Similarly \(I \propto 2\lambda\), \(I \propto 3\lambda\), \(I \propto 4\lambda\). |
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