The value of int_0^{pi /2} frac{dx}{1 + tan^3 | vedclass

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(D) Let $I = \int_0^{\pi /2} \frac{dx}{1 + 	an^3 x}$.Using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$,we have:$I = \int_0^{\pi /2} \frac{dx

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$\frac{\pi}{4}$

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(D) Let $I = \int_0^{\pi /2} \frac{dx}{1 + 	an^3 x}$.Using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$,we have:$I = \int_0^{\pi /2} \frac{dx}{1 + 	an^3(\pi/2 - x)} = \int_0^{\pi /2} \frac{dx}{1 + \cot^3 x}$.$I = \int_0^{\pi /2} \frac{	an^3 x}{1 + 	an^3 x} dx$.Adding the two expressions for $I$:$2I = \int_0^{\pi /2} \left( \frac{1}{1 + 	an^3 x} + \frac{	an^3 x}{1 + 	an^3 x} ight) dx$.$2I = \int_0^{\pi /2} \frac{1 + 	an^3 x}{1 + 	an^3 x} dx = \int_0^{\pi /2} 1 dx$.$2I = [x]_0^{\pi /2} = \frac{\pi}{2}$.Therefore,$I = \frac{\pi}{4}$.

The value of $\int_0^{\pi /2} \frac{dx}{1 + \tan^3 x}$ is

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