Chapter 7.2, Problem 39E

To determine

To evaluate the integral .

Answer to Problem 39E

The value of the ${\displaystyle \int \frac{dx}{x\ln x}}=\ln \left|\ln x\right|+C$

Explanation of Solution

Given information:

The integral is ${\displaystyle \int \frac{dx}{x\ln x}}$

Calculation:

The integral is ${\displaystyle \int \frac{dx}{x\ln x}}$

Using substitution method of integration

  $\begin{array}{l}{\displaystyle \int \frac{dx}{x\ln x}}\\ \text{let }\ln x=t\\ \text{differentiating both side}\\ \frac{1}{x}dx=dt\\ dx=xdt\\ {\displaystyle \int \frac{dx}{x\ln x}}\\ {\displaystyle \int \frac{1}{xt}}\times xdt\\ {\displaystyle \int \frac{1}{t}}dt=\ln \left|t\right|+C=\ln \left|\ln x\right|+C,\text{where }C\text{ is constant}\text{.}\\ \end{array}$

Hence ,the value of the ${\displaystyle \int \frac{dx}{x\ln x}}=\ln \left|\ln x\right|+C$