Chapter 7.2, Problem 33E

To determine

To evaluate the integral .

Answer to Problem 33E

The value of the ${\displaystyle \int \frac{{\ln }^{6}x}{x}}dx=\frac{{\ln }^{7}x}{7}+C$

Explanation of Solution

Given information:

The integral is ${\displaystyle \int \frac{{\ln }^{6}x}{x}}dx$

Calculation:

The integral is ${\displaystyle \int \frac{{\ln }^{6}x}{x}}dx$

Using substitution method of integration

  $\begin{array}{l}{\displaystyle \int \frac{{\ln }^{6}x}{x}}dx\\ \text{let }\ln x=t\\ \text{differentiating both side}\\ \frac{1}{x}dx=dt\\ dx=xdt\\ {\displaystyle \int \frac{t^6}{x}}\times xdt\\ {\displaystyle \int t^6}dt=\frac{t^7}{7}+C=\frac{{\ln }^{7}x}{7}+C,\text{ where }C\text{ is constant of integration}\text{.}\end{array}$

Hence ,the value of the ${\displaystyle \int \frac{{\ln }^{6}x}{x}}dx=\frac{{\ln }^{7}x}{7}+C$ .