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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let $[\cdot]$ be the greatest integer function. If $\alpha=\int_0^{64}(x^{1/3}-[x^{1/3}])dx$, then $\dfrac{1}{\pi}\int_0^{\alpha\pi}\left(\dfrac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\right)d\theta$ is equal to ____ (Integer Type)
Solution
Split integral over $[k^3,(k+1)^3]$. Evaluate $\alpha=\sum_{k=0}^{3}\int_{k^3}^{(k+1)^3}(x^{1/3}-k)dx=4$. Then integral becomes periodic and evaluates to $\dfrac{1}{\pi}\cdot4\cdot\dfrac{\pi}{2}=2$.Online Test Series, Information About Examination,
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