Let [ · ] be the greatest integer function. If = 0 64 x 1 / 3 - [ x 1 / 3 ] d x , then 1 0 ( sin 2 sin 6 + cos 6 ) ? d is equal to_______. [2026]

Question

Let $[\cdot]$ be the greatest integer function. If $α = \int_{0}^{64} (x^{1 / 3} - [x^{1 / 3}]) d x,$ then $\frac{1}{π} \int_{0}^{α π} (\frac{\sin^{2} θ}{\sin^{6} θ + \cos^{6} θ}) d θ$ is equal to_______. [2026]

Smart Achievers · Accepted Answer

(36)

$∵$ $\int_{0}^{64} x^{\frac{1}{3}} d x = \frac{3}{4} {[x^{\frac{4}{3}}]}_{0}^{64} = 192$

and

$\int_{0}^{64} [x^{1 / 3}] d x = \int_{0}^{1} [x^{1 / 3}] d x + \int_{1}^{8} [x^{1 / 3}] d x + \int_{8}^{27} [x^{1 / 3}] d x + \dots + \int_{27}^{64} [x^{1 / 3}] d x = 156$

$So α = 192 - 156 = 36$

Now

$E = \frac{1}{π} \int_{0}^{36 π} \frac{\sin^{2} θ}{\sin^{6} θ + \cos^{6} θ} d θ$

$= \frac{36}{π} \int_{0}^{π} \frac{\sin^{2} θ}{\sin^{6} θ + \cos^{6} θ} d θ$

$\Rightarrow E = \frac{36 \cdot 2}{π} \int_{0}^{π / 2} \frac{\sin^{2} θ}{\sin^{6} θ + \cos^{6} θ} d θ$

Let $J = \int_{0}^{π / 2} \frac{\sin^{2} θ}{\sin^{6} θ + \cos^{6} θ} d θ . . . (1)$

Applying King property,

$J = \int_{0}^{π / 2} \frac{\cos^{2} θ}{\sin^{6} θ + \cos^{6} θ} d θ . . . (2)$

Now

$2 J = \int_{0}^{π / 2} \frac{1}{\sin^{6} θ + \cos^{6} θ} d θ (add (1) & (2))$

$= \int_{0}^{π / 2} \frac{\sec^{6} θ}{\tan^{6} θ + 1} d θ$

$= \int_{0}^{\infty} \frac{1 + λ^{2}}{λ^{4} - λ^{2} + 1} d λ$

$= \int_{0}^{\infty} \frac{1 + \frac{1}{λ^{2}}}{λ^{2} - 1 + \frac{1}{λ^{2}}} d λ$

$= π$

$\Rightarrow J = \frac{π}{2}$

$\Rightarrow E = \frac{36 \cdot 2}{π} \times J = 36$

Solution

Q.
Let $[\cdot]$ be the greatest integer function. If $α = \int_{0}^{64} (x^{1 / 3} - [x^{1 / 3}]) d x,$ then $\frac{1}{π} \int_{0}^{α π} (\frac{\sin^{2} θ}{\sin^{6} θ + \cos^{6} θ}) d θ$ is equal to_______. [2026]

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Solution

Q. Let [·] be the greatest integer function. If α=∫064(x1/3-[x1/3]) dx, then 1π∫0απ(sin2θsin6θ+cos6θ) dθ is equal to_______. [2026]

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Q.
Let $[\cdot]$ be the greatest integer function. If $α = \int_{0}^{64} (x^{1 / 3} - [x^{1 / 3}]) d x,$ then $\frac{1}{π} \int_{0}^{α π} (\frac{\sin^{2} θ}{\sin^{6} θ + \cos^{6} θ}) d θ$ is equal to_______. [2026]