complex variable

Exercise 1. Evaluate the real improper integral

Z +1

�1

dx

x4 + 1 .

Exercise 2. Evaluate the integral

Z +1

0

dx

x5 + 1 .

Exercise 3. For a given constant a > 0, evaluate the integral Z 1

0

cos ax

(x2 + 9)2 dx.

Exercise 4. Evaluate the integral

Z +1

0

dx p x(x4 + 1)

Exercise 5. Evaluate the integral

Z 2⇡

0

d✓

5 + 4 cos 2✓ .

Exercise 6. Determine the number of roots, counting multiplicities, of the equation

z 9 � 2z2 + 11z + 1 = 0

in the annulus 1  |z| < 2.