We are to evaluate, int x/(x + 4)^(1/2)*dx∫x(x+4)12⋅dx.
We shall accomplish this by substitution.
Let, (x + 4) = t(x+4)=t
implies dx = dt⇒dx=dt (By simple differentiation).
Thus, the integral becomes,
int x/(x +4)^(1/2)*dx = int (t - 4)/t^(1/2)*dt∫x(x+4)12⋅dx=∫t−4t12⋅dt
= int t^(1/2)*dt - int 4t^(-1/2)*dt=∫t12⋅dt−∫4t−12⋅dt
=2/3 t^(3/2) - 8t^(1/2) + C=23t32−8t12+C, where CC is the integration constant.
In terms of xx, the integral may be now written as,
int x/(x + 4)^(1/2)*dx =2/3 (x + 4)^(3/2) - 8(x + 4)^(1/2) + C∫x(x+4)12⋅dx=23(x+4)32−8(x+4)12+C.
