x*(arctan根号下x)的微积分

Question

x*(arctan 根号下x) 的微积分

Answer 1

答：
设t=√x,x=t^2
∫ xarctan√x dx
=∫ (t^2) arctan t d(t^2)
=2 ∫ (t^3) arctant dt
=(1/2) ∫ arctant d(t^4)
=(1/2)*(t^4)*arctant-(1/2)∫ (t^4) d(arctant)
=(1/2)*(t^4)*arctant-(1/2)∫ [ (t^2+1-1)^2] /(1+t^2) dt
=(1/2)*(t^4)*arctant-(1/2) ∫ [ t^2+1-2+1/(1+t^2) ] dt
=(1/2)*(t^4)*arctant-(1/2) * [(1/3)t^3 - t +arctant ]+C
=(1/2)(t^4 -2) *arctant + t/2 -(1/6)*(t^3)+C
=(1/2)*(x^2 -2) *arctan√x+(√x)/2-(1/6)*x√x+C