intlimits_{frac{-pi}{2}}^{frac{pi}{2}} frac{x | vedclass

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Question

(C) माना $I = \int\limits_{-\pi/2}^{\pi/2} \frac{x^2}{1 + 	an x + \sec x} \, dx$ (क्योंकि $x \in [-\pi/2, \pi/2]$ के लिए $\sqrt{1+	an^2 x} = |\sec x

Vedclass · Accepted Answer

$\frac{\pi^3}{24}$

Vedclass · Answer

(C) माना $I = \int\limits_{-\pi/2}^{\pi/2} \frac{x^2}{1 + 	an x + \sec x} \, dx$ (क्योंकि $x \in [-\pi/2, \pi/2]$ के लिए $\sqrt{1+	an^2 x} = |\sec x| = \sec x$ होता है)।गुणधर्म $\int_{-a}^{a} f(x) \, dx = \int_{0}^{a} (f(x) + f(-x)) \, dx$ का उपयोग करने पर:$I = \int_{0}^{\pi/2} \left( \frac{x^2}{1 + 	an x + \sec x} + \frac{(-x)^2}{1 + 	an(-x) + \sec(-x)} ight) \, dx$$I = \int_{0}^{\pi/2} x^2 \left( \frac{1}{1 + 	an x + \sec x} + \frac{1}{1 - 	an x + \sec x} ight) \, dx$$I = \int_{0}^{\pi/2} x^2 \left( \frac{1 + \sec x - 	an x + 1 + \sec x + 	an x}{(1 + \sec x)^2 - 	an^2 x} ight) \, dx$$I = \int_{0}^{\pi/2} x^2 \left( \frac{2 + 2\sec x}{1 + 2\sec x + \sec^2 x - 	an^2 x} ight) \, dx$चूंकि $1 + 	an^2 x = \sec^2 x$,इसलिए हर $1 + 2\sec x + 1 = 2 + 2\sec x$ हो जाता है।$I = \int_{0}^{\pi/2} x^2 \left( \frac{2(1 + \sec x)}{2(1 + \sec x)} ight) \, dx = \int_{0}^{\pi/2} x^2 \, dx$$I = \left[ \frac{x^3}{3} ight]_{0}^{\pi/2} = \frac{(\pi/2)^3}{3} = \frac{\pi^3}{8 	imes 3} = \frac{\pi^3}{24}$.

$\int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{x^2}{1 + \tan x + \sqrt{1 + \tan^2 x}} \, dx$ का मान ज्ञात कीजिए।

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