求∫dx/[x^(1/2)+x^(1/4)]dx

Question

求∫dx/[x^(1/2)+x^(1/4)]dx

Answer 1

设x = t^4
那么dx = 4t^3dt
原式 =4 ∫ t^3 /(t^2+t) dt = 4 ∫ t^2 / (t+1) dt = 4 ∫ (t - 1/(1+t) )dt = 2t^2 - 4ln(1+t) + C = 2x^(1/2) - 4ln(1+x^(1/4))+C