Use integration by parts to evaluate the integral.
∫ from 1 to 4 of (√x) (ln√x) dx
Im not really sure what to choose for u and dv.
Thank you so much for the help!!
Try to set u to something that gets simpler in the derivative.
u = ln √x
du = 1/√x * 1/2√x dx = dx/2x
dv = √x
v = 2/3 x3/2
∫ u dv = uv - ∫v du
= 2/3 x3/2 ln √x - ∫ 1/3 √x dx
= 2/3 x3/2 ln √x - 2/9 x3/2
= 2/9 x3/2 (3 ln √x - 1)
F(4) = 16/9 (ln 8 - 1)
F(1) = -2/9
F(4) - F(1) = 16/9 ln 8 - 14/9