Chapter 9.4, Problem 42E

To determine

Tofind: the value of integral and state that it is diverges or not.

Answer to Problem 42E

The integral is diverges.

Explanation of Solution

Given information:

The integral is ${\displaystyle {\int }_{-1}^1\ln \left(\left|w\right|\right)dw}$ .

Calculation: to find the value of integral, first integrate then simplify,

  $\begin{array}{l}{\displaystyle {\int }_{-1}^1\ln \left(\left|w\right|\right)dw}=2{\displaystyle {\int }_0^1\ln \left(w\right)dw}\text{ [function is even]}\\ {\displaystyle {\int }_{-1}^1\ln \left(\left|w\right|\right)dw}=2{\left[w\left(\ln w-1\right)\right]}_0^2\text{ [ }{\displaystyle \int uvdx=u{\displaystyle \int vdx-{\displaystyle \int \frac{du}{dx}{\displaystyle \int vdxdx}}}}]\\ {\displaystyle {\int }_{-1}^1\ln \left(\left|w\right|\right)dw}=2[2\left(\ln 2-1\right)-0\left(\ln 0-1\right)]\end{array}$

Thus, the integral is diverges.