Problem 2 - Entrance Test

Use integration by parts: u = ln x, dv = x^3 dx ⇒ du = dx/x, v = x^4/4. The integral becomes [x^4 ln x / 4]_0^1 - ∫0^1 x^3/4 dx. The boundary term is 0 (limit as x→0+ of x^4 ln x = 0) and the remaining integral is -1/4 · [x^4/4]_0^1 = -1/16.