int_{-pi / 2}^{pi / 2} frac{cos ^{2} x}{1+3^{ | vedclass

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(A) माना $I = \int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} dx$  --- $(1)$गुणधर्म $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$ का उपयोग क

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$\frac{\pi}{4}$

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(A) माना $I = \int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} dx$  --- $(1)$गुणधर्म $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$ का उपयोग करने पर:$I = \int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2}(-x)}{1+3^{-x}} dx = \int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+\frac{1}{3^{x}}} dx = \int_{-\pi / 2}^{\pi / 2} \frac{3^{x} \cos ^{2} x}{3^{x}+1} dx$ --- $(2)$$(1)$ और $(2)$ को जोड़ने पर:$2I = \int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x + 3^{x} \cos ^{2} x}{1+3^{x}} dx$$2I = \int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x(1+3^{x})}{1+3^{x}} dx = \int_{-\pi / 2}^{\pi / 2} \cos ^{2} x dx$चूँकि $\cos^{2} x$ एक सम फलन है:$2I = 2 \int_{0}^{\pi / 2} \cos ^{2} x dx$$I = \int_{0}^{\pi / 2} \frac{1+\cos 2x}{2} dx = \frac{1}{2} [x + \frac{\sin 2x}{2}]_{0}^{\pi / 2}$$I = \frac{1}{2} [(\frac{\pi}{2} + 0) - (0 + 0)] = \frac{\pi}{4}$

$\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} d x$ का मान ज्ञात कीजिए।

Similar Questions

निश्चित समाकलनों के गुणों का उपयोग करके,$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x \, dx$ का मान ज्ञात कीजिए।

$\int_{-\pi}^{\pi} x^2 \sin x \, dx =$

यदि $b = \int_{0}^{1} \frac{e^{t}}{t+1} dt$ है,तो $\int_{a-1}^{a} \frac{e^{-t}}{t-a-1} dt$ का मान ज्ञात कीजिए।

$\int_{-1}^3 \left(\tan^{-1}\left(\frac{x}{x^2+1}\right) + \tan^{-1}\left(\frac{x^2+1}{x}\right)\right) dx =$

यदि $I = \int_0^{\frac{\pi}{2}} \cos(\sin x) \,dx$,$J = \int_0^{\frac{\pi}{2}} \sin(\cos x) \,dx$,और $K = \int_0^{\frac{\pi}{2}} \cos x \,dx$ है,तो:

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