# 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.

###### This problem is original. The picture of the graph was produced from Desmos.

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