# 3.2 Further exercises

Here are some further exercises to end this section.

## Exercise 14

For each of the following functions , evaluate

where is any contour from to .

### Answer

In each case, is continuous on and has a primitive on , so we can apply the Fundamental Theorem of Calculus to evaluate the integral using any contour from to .

7. A primitive of is

   Hence

8. A primitive of is

   Hence

9. Let , . Then and are entire (that is, and are differentiable on the whole of ), and and are entire and hence continuous. Then, using Integration by Parts (Theorem 9), we have

## Exercise 15

Evaluate the following integrals. (In each case pay special attention to the hypotheses of the theorems you use.)

1. ,

   where is the arc of the circle from to passing through 1.

2. , where is as in part (a).

3. , where is the unit circle .

4. , where is the circle .

(Hint: For part (c), use the identity .)

### Answer

1. Let , and . Then is continuous on , is a primitive of on , and is a contour in . Thus, by the Fundamental Theorem of Calculus,

2. Let , and . Then is continuous on , is a primitive of on , and is a contour in . Thus, by the Fundamental Theorem of Calculus,

3. The function

   is continuous and has an entire primitive . Thus, by the Closed Contour Theorem,

4. Let , and . Then is continuous on , is a primitive of on , and is a contour in . Thus, by the Closed Contour Theorem,

## Exercise 16

Construct a grid path from to in the domain of the function , for each of the following cases.

1. ,

2. ,

### Answer

The domain of is the region

1. The figure shows one grid path in from 1 to 6 (there are many others).

2. The figure shows one grid path in from to (again, there are many others).

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