Question

Solve the integral:$\int \ln \left(3x\right)dx$

Short answer

The required answer is$x\left(\ln \left(3x\right)-1\right)+c$.

Step-by-step solution

Step 1. Given information. 

We have given integral is$\int \ln \left(3x\right)dx$.

Step 2. Solve the integration by parts .  

We have,

$$\begin{gathered} u=\ln \left(3x\right) \\ du=\frac{dx}{x} \end{gathered}$$

and

$$\begin{gathered} dv=dx \\ v=\int dx \\ v=x \end{gathered}$$

The formula of integration by parts is $\int udv=uv-\int vdu$.

$$\begin{gathered} \int \ln \left(3x\right)dx \\ =\left(\ln \left(3x\right)x-\int 1dx\right) \\ =x\ln \left(3x\right)-x+c \\ =x\left(\ln \left(3x\right)-1\right)+c \end{gathered}$$