Does the improper integral
exist? If yes, what is its value?
What is its Cauchy principal value?
Does the integral
have a finite Cauchy principal value?
Reminder
The definition of the improper integral of case 3, with a singular point of the function at x0 with
is
(2.2.3.19)
One should notice that the definition of case 3, as it is expressed by (2.2.3.19), requires two independent convergences for the same singular point.
... converging the two expressions by a single limit:
(2.2.3.21)
... could it happen that the limit exists, while the improper integral does not? As matter of fact, yes! In such a case the limit is called the Cauchy principal value of the integral apearing on the left of (2.2.3.21).
Parts 1-5
Solution of question 1.
The corresponding indefinite integral is
Within the interval of integration, tanx is singular at point
therefore, according to (2.2.3.19) the required improper integral is
After substituting the indefinite integral of part 1, we obtain
Each one of the limits diverges to infinity, therefore the improper integral does not exist.
Parts 6-8
Solution of question 2
For the Cauchy principal value of the integral, one have to use the same ε for both limits of part 4
Part 9
Solution of question 3.
The singular point of |tanx| diverges from the left and from the right to positive infinity. Therefore applying the same ε for both expressions cannot cancel the infinities.
Score
By parts:
Parts 1,2,3,4,5,6,7,8 are worth 1 point each.
Part 9 is worth 2 points
By questions:
Question 1 is worth 5 points.
Question 2 is worth 3 points.
Question 3 is worth 2 points.