Question

Solve the integral:$\int \ln \sqrt{x}dx$

Short answer

The required answer is$\frac{x}{2}\ln \left(x\right)-\frac{x}{2}+c$.

Step-by-step solution

Step 1. Given information. 

We have given integral is$\int \ln \sqrt{x}dx$.

Step 2. Solve the integration by parts . 

We have,

$$\begin{gathered} u=\ln x \\ du=\frac{dx}{x} \end{gathered}$$

and

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

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

$$\begin{gathered} \int \ln \sqrt{x}dx \\ =\int \left(\frac{1}{2}\ln x\right)dx \\ =\frac{1}{2}\int \ln xdx \\ =\frac{1}{2}\left(\ln \left(x\right)x-\int x.\frac{1}{x}dx\right) \\ =\frac{1}{2}\left(\ln \left(x\right)x-\int 1dx\right) \\ =\frac{x}{2}\ln \left(x\right)-\frac{x}{2}+c \end{gathered}$$