Integration with Square Roots

Calculus Level 3

Given that $\displaystyle \int_0^4 x^3\sqrt{9+x^2} dx = a$ , what is the value of $\lfloor a \rfloor$ ?

Details and assumptions

Greatest Integer Function: $\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}$ refers to the greatest integer less than or equal to $x$ . For example $\lfloor 2.3 \rfloor = 2$ and $\lfloor -3.4 \rfloor = -4$ .