Evaluate the integral. (Use C for the constant of integration.)
integral of ln(3x+1)dx
To integrate ln(3x+1)dx, we can use integration by substitution.
Let u = 3x+1, then du = 3dx.
Rearranging this equation, we have dx = du/3.
Now substitute these values into the integral:
∫ln(3x+1)dx = ∫ln(u) (du/3)
Using the property of logarithms, ln(u) = ln|u| + C.
= (1/3) ∫ln|u|du
Now substitute back u = 3x+1:
(1/3) ∫ln|3x+1|du
= (1/3) ∫ln|3x+1| (3dx)
= (1/3) ∫ln|3x+1| (3dx)
= ∫ln|3x+1|dx
Therefore, the integral of ln(3x+1)dx is ∫ln|3x+1|dx + C.