# Infinite dimensions, not quite infinite volume

Calculus Level 5

$\lim _{ x\rightarrow \infty }{ { \left( 1+6\int _{ 0 }^{ B }{ \frac { y{ \left( x+{ y }^{ 2 } \right) }^{ 2 } }{ { x }^{ 3 } } dy } \right) }^{ x } } ={ e }^{ 2016 }$ The real number $$B$$ satisfies the equation above and can be expressed in the form $$\alpha \sqrt { \beta }$$, with $$\alpha$$ and $$\beta$$ positive, root-free integers with a greatest common divisor of 2. Find $$\alpha+\beta$$.

If you believe that the following limit goes to infinity for all real $$B$$, submit your answer as 2016.

Details and Assumptions

$$e$$ denotes Euler's number, the base of the natural logarithm.

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