સંકલન int limits_{-pi / 2}^{pi / 2} frac{d x} | vedclass

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Question

(D) ધારો કે $I = \int \limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}$.ગુણધર્મ $\int_{a}^{b} f(x) dx

Vedclass · Accepted Answer

$\frac{\pi}{2}$

Vedclass · Answer

(D) ધારો કે $I = \int \limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}ight)\left(\sin ^{6} x+\cos ^{6} xight)}$.ગુણધર્મ $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$ નો ઉપયોગ કરતા,આપણને મળે છે $I = \int \limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{-x}ight)\left(\sin ^{6} x+\cos ^{6} xight)}$.$I$ માટેના બંને પદોનો સરવાળો કરતા:$2I = \int \limits_{-\pi / 2}^{\pi / 2} \left( \frac{1}{1+e^x} + \frac{1}{1+e^{-x}} ight) \frac{dx}{\sin^6 x + \cos^6 x}$.કારણ કે $\frac{1}{1+e^x} + \frac{1}{1+e^{-x}} = 1$,તેથી:$2I = \int \limits_{-\pi / 2}^{\pi / 2} \frac{dx}{\sin^6 x + \cos^6 x}$.વિધેય યુગ્મ હોવાથી,$2I = 2 \int \limits_{0}^{\pi / 2} \frac{dx}{\sin^6 x + \cos^6 x}$,તેથી $I = \int \limits_{0}^{\pi / 2} \frac{dx}{\sin^6 x + \cos^6 x}$.અંશ અને છેદને $\cos^6 x$ વડે ભાગતા:$I = \int \limits_{0}^{\pi / 2} \frac{\sec^6 x dx}{	an^6 x + 1} = \int \limits_{0}^{\pi / 2} \frac{(1+	an^2 x)^2 \sec^2 x dx}{	an^6 x + 1}$.ધારો કે $	an x = t$,તો $\sec^2 x dx = dt$:$I = \int \limits_{0}^{\infty} \frac{(1+t^2)^2}{t^6+1} dt = \int \limits_{0}^{\infty} \frac{1+2t^2+t^4}{t^6+1} dt$.આ સંકલનનું મૂલ્ય $\frac{\pi}{2}$ મળે છે.

સંકલન $\int \limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}$ નું મૂલ્ય શોધો.

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