Stuffed with everything I could think of

Calculus Level pending

\[\large{f\left( x \right) =\left| \begin{matrix} { sin }^{ 5 }x\quad & ln\left( sin(x) \right) \quad & \frac { \sqrt { sin(x) } }{ \sqrt { sin(x) } +\sqrt { cos(x) } } \\ \left\lceil n \right\rceil & \left\lfloor \sum _{ k=1 }^{ 10000 }{ k } \right\rfloor & \left\{ \prod _{ k=1 }^{ 10000 }{ k } \right\} \\ \frac { 8 }{ 15 } & \frac { \pi }{ 2 } ln\left( \frac { 1 }{ 2 } \right) & \frac { \pi }{ 4 } \end{matrix} \right| } \]

\[\large{\therefore \quad \quad \quad \left\{ \int _{ 0 }^{ \frac { \pi }{ 2 } }{ f\left( x \right) } dx \right\} =?} \]

\(\left\lceil . \right\rceil \quad and\quad \left\lfloor . \right\rfloor \) represents the ceiling and floor function respectively.

\(\left\{ . \right\} \) is the fractional part function.

\(\sum { } \quad and\quad \prod { } \) are sum and product functions respectively.


This is a problem of my set JEE Calculus.
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