int_{0}^{frac{pi}{2}} frac{sin ^{4} x}{sin ^{ | vedclass

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(B) ધારો કે $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x}{\sin ^{4} x+\cos ^{4} x} d x$ ..... $(1)$ગુણધર્મ $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(B) ધારો કે $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x}{\sin ^{4} x+\cos ^{4} x} d x$ ..... $(1)$ગુણધર્મ $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ નો ઉપયોગ કરતા:$I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} (\frac{\pi}{2} - x)}{\sin ^{4} (\frac{\pi}{2} - x) + \cos ^{4} (\frac{\pi}{2} - x)} dx$કારણ કે $\sin(\frac{\pi}{2} - x) = \cos x$ અને $\cos(\frac{\pi}{2} - x) = \sin x$,તેથી:$I = \int_{0}^{\frac{\pi}{2}} \frac{\cos ^{4} x}{\cos ^{4} x + \sin ^{4} x} dx$ ..... $(2)$$(1)$ અને $(2)$ નો સરવાળો કરતા:$2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x + \cos ^{4} x}{\sin ^{4} x + \cos ^{4} x} dx$$2I = \int_{0}^{\frac{\pi}{2}} 1 dx = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2}$તેથી,$I = \frac{\pi}{4}$.

$\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x}{\sin ^{4} x+\cos ^{4} x} d x$ ની કિંમત શોધો.

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