1 True or False (1) f(x)= ) 1,xis rational 0,xis irrational is integrable on any bounded closed interval [a,b] (2)The derivative of y=y(x) determined by the formula arctan(y)/(x)=lnsqrt (x^2+y^2) is y'=(x+y)/(x-y)

Let's analyze each statement one by one:(1) The function is defined as for rational and for irrational . This function is not integrable on any bounded closed interval because it is not continuous. The function jumps between values of 0 and 1 at every irrational point in the interval, which means it has an infinite number of discontinuities. Therefore, the statement is False.(2) To find the derivative of determined by the equation , we need to differentiate both sides of the equation with respect to . Let's differentiate the left side: Now, let's differentiate the right side: Equating the derivatives from both sides, we get: Multiplying both sides by , we have: Simplifying, we get: Therefore, the statement is True.