No computational aids required!

Calculus Level 5

Let \(f : [0,1] \to \mathbb{R} \) be a function defined as follows: \[\large f(x) = 10 x^{\sin x + \cos x} \]

Enter the value of the following expression:

\[ \left \lfloor \int_0^1 f(x) \, dx\right \rfloor\]

Details and Assumptions:

\(\lfloor \cdot \rfloor\) denotes the floor function.
This question is meant to be done without computational aids.

Source: CMI enterance examination.