∫xdx/(1+根号下x)

Question

∫xdx/(1+根号下x)

Answer 1

令t = √x,x = t²,dx = 2tdt
∫ xdx/(1 + √x) = ∫ t²(2tdt)/(1 + t)
= 2∫ t³/(1 + t) dt
= 2∫ [(t³ + 1) - 1]/(t + 1) dt
= 2∫ [(t + 1)(t² - t + 1) - 1]/(t + 1) dt
= 2∫ (t² - t + 1) dt - 2∫ dt/(t + 1)
= 2(t³/3 - t²/2 + t) - 2ln|1 + t| + C
= (2/3)t³ - t² + 2t - 2ln|1 + t| + C
= (2/3)x^(3/2) - x + 2√x - 2ln|1 + √x| + C