\[\large \displaystyle \dfrac { da }{ db } ={ a }^{ 2 }+\int _{ 0 }^{ { \pi } / { 4 } }{ \left( { a }^{ 2 } \right) \, db } \]

For the differential equation above, \(a=f(b)\) and \(a(0)=0\). Find \(\left\lceil a\left( 0.5 \right) \right\rceil \).

**Details and Assumptions:**

Use radians only.

You may use a calculator to do some heavy calculations.

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