Greatest Integer Function Integral (Jee Mains)

Definite Integration Properties

Hard

Greatest Integer Function Integral

Let [•] be the greatest integer function. If $\alpha = \int_0^{64} {\left( {{x^{1/3}} - \left[ {{x^{1/3}}} \right]} \right)dx}$ , then $\frac{1}{\pi }\int_0^{\alpha \pi } {\left( {\frac{{{{\sin }^2}\theta }}{{{{\sin }^6}\theta + {{\cos }^6}\theta }}} \right)} \,d\theta$ is equal to ____.

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