Calculus Help Please! Difficult and need thinking

Question

If f is integrable on [a, b], then f2 is integrable on [a, b]. This statement however is false for the improper integral. Show it by an example. ( Please Explain each step )

Answer 1

Consider the function f(x) = x^-1/2 . The improper integral over [0,1] evaluates to:

∫₀¹ x^-1/2dx = [2√x]₀¹ = 2

However, we have f²(x) = 1/x, which results in:

∫₀¹ x^-1dx = [ln x]₀¹ = 0 - ln 0 = ∞

Therefore, f is integrable but f² is not integrable.