int_0^{pi / 2} frac{d x}{1+tan ^3 x} ની કિંમત | vedclass

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(C) ધારો કે $I = \int_0^{\pi / 2} \frac{d x}{1+	an ^3 x}$.$I = \int_0^{\pi / 2} \frac{\cos ^3 x}{\sin ^3 x+\cos ^3 x} d x$ ...$(i)$ગુણધર્મ $\int_0^a

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$\frac{\pi}{4}$

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(C) ધારો કે $I = \int_0^{\pi / 2} \frac{d x}{1+	an ^3 x}$.$I = \int_0^{\pi / 2} \frac{\cos ^3 x}{\sin ^3 x+\cos ^3 x} d x$ ...$(i)$ગુણધર્મ $\int_0^a f(x) d x = \int_0^a f(a-x) d x$ નો ઉપયોગ કરતા:$I = \int_0^{\pi / 2} \frac{\cos ^3(\frac{\pi}{2}-x)}{\sin ^3(\frac{\pi}{2}-x)+\cos ^3(\frac{\pi}{2}-x)} d x$$I = \int_0^{\pi / 2} \frac{\sin ^3 x}{\cos ^3 x+\sin ^3 x} d x$ ...(ii)સમીકરણ $(i)$ અને (ii) નો સરવાળો કરતા:$2I = \int_0^{\pi / 2} \frac{\cos ^3 x + \sin ^3 x}{\sin ^3 x+\cos ^3 x} d x$$2I = \int_0^{\pi / 2} 1 d x = [x]_0^{\pi / 2} = \frac{\pi}{2}$.તેથી,$I = \frac{\pi}{4}$.

$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ ની કિંમત શોધો:

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