Determine whether the following functions are Riemann integrable on [0, 1] or not.(1) f (x) := sin √x√1 + x2

The function f(x) = sin(√x)√(1 + x^2) is Riemann integrable on the interval [0, 1].

Here's why:

1. The function is defined and continuous on the interval [0, 1]. The square root function, sine function, and polynomial function are all continuous functions. The composition of continuous functions is also continuous. Therefore, f(x) is continuous on [0, 1].

2. The function is bounded on the interval [0, 1]. This is because the sine function is always between -1 and 1, and the square root function and polynomial function are always non-negative. Therefore, the product of these functions is also bounded.

According to the Riemann integrability criterion, a function is Riemann integrable on an interval [a, b] if and only if it is bounded and continuous on that interval. Therefore, f(x) = sin(√x)√(1 + x^2) is Riemann integrable on [0, 1].