积分∫x*sqrt(ax+b)dx=2/15a^2*(3ax-2b)*sqrt[(ax+b)^3]+c

Question

积分∫x*sqrt(ax+b)dx=2/15a^2*(3ax-2b)*sqrt[(ax+b)^3]+c

Answer 1

∫x√(ax+b)dx
=1/a ∫x√(ax+b)d(ax+b)
=2/3a ∫xd(ax+b)^(3/2)
=2/(3a)[x(ax+b)^(3/2)-∫(ax+b)^(3/2)dx]
=2/3a*x(ax+b)^(3/2)-2/(3a^2)∫(ax+b)^(3/2)d(ax+b)
=2/(3a)*x(ax+b)^(3/2)-4/(15a^2)*(ax+b)^(5/2)+C