Example 8.13.      does not exist,     and     .

Explore Solution 8.13.

Solution.

If we attempt to use Equation (8-7) then we obtain

and the last computation   ""   is undefined .

Thus, the improper integral      does not exist.

If we use Equation (8-8) then we obtain

This computation is well defined and is known as the Cauchy principal value () of   .

Therefore,

.

The area      "cancels out" the area   .

Here we can see the value of the integrals, and that    .

We are done.

Aside.  Both and can be used to investigate the integrals.

Aside.  We can let Mathematica compute the improper integral.

If we attempt to use Equation (8-7) then we obtain

``````

```

```

If we use Equation (8-8) then we obtain

``````

```

```

```

```

```

```

Aside.  Mathematica Vers. 7 and Vers. 8 can find the Principal Value of the integral.

``````

``````

We are really done.

Aside.  We can let Maple compute the improper integral.

If we attempt to use Equation (8-7) then we obtain

>

If we use Equation (8-8) then we obtain

>

>

>

Aside.  Maple 12 can find the Cauchy Principal Value of the integral.

>

Remark.  In this book the use of computers is optional.

Hopefully this text will promote their use and understanding.

(c) 2012 John H. Mathews, Russell W. Howell