Question

What is $\int \frac{\ln x}{x}dx$ equal to ?

• A. $\frac{{\left(\ln x\right)}^2}{2}+c$ where $c$ is the constant of integration
• B. $\frac{\left(\ln x\right)}{2}+c$ where $c$ is the constant of integration
• C. ${\left(\ln x\right)}^2+c$ where $c$ is the constant of integration
• D. None of the above

Answer 1

Answer: (A)

$\frac{{\left(\ln x\right)}^2}{2}+c$ where $c$ is the constant of integration

Let $I=\int \frac{\ln x}{x}dx$

Put $\ln x=t$ $\Rightarrow \frac{dx}{x}=dt$

Then, we get

$I=\int tdt=\frac{t^2}{2}+c=\frac{{\left(\ln x\right)}^2}{2}+c$

Hence, A is correct.