(0, (1/n, if x = m/n in lowest terms, n>0. if x is irrational or 0 E. Let f(x)= (a) Show that f is Riemann integrable on [0, 1]. (b) Find Sf.

Evaluate the definite intagral  from  3 to 0   4(t^5-t^3+1)dt

Could explain how to do this in detail?

Let \( g \) be a bounded function on [0,1] and assume that the restriction of \( g \) to the interval [1/n,1] is Riemann integrable for every \( n \geq 2 \). Show that \( g \) is Riemann integrable over the entire interval [0,1]. (Hint: Let \( \epsilon > 0 \) be given and let \( M > 0 \) be a constant such that \( |g(x)| \leq M \) for all \( x \) in [0,1]. Choose \( n \geq 2 \) such that \( \frac{1}{n} < \frac{\epsilon}{3M} \) and note that $$ \sup\{g(x): x \in [0,1/n]\}1/n - \inf\{g(x): x \in [0,1/n]\}1/n < \frac{2\epsilon}{3} $$ Now use that \( g \) is Riemann integrable on [1/n,1] to find an appropriate partition of [0,1].)