Final answer:
To integrate f(z) counterclockwise over the unit circle, we need to write it in terms of partial fractions. By factoring the denominator and finding the common denominator, we can express f(z) as (1/2)/z + (-1/2)/(z+2). We can then integrate this expression using the parameterization z = e^(it).
Step-by-step explanation:
To write f(z) in terms of partial fractions and integrate it counterclockwise over the unit circle, we first need to factor the denominator of the fraction, which is z^2 + 2z. We can rewrite this as z(z+2). Since the numerator is z+1, we can express f(z) as:
f(z) = (z+1)/(z(z+2))
Next, we need to find the partial fraction decomposition. We separate f(z) into two fractions:
f(z) = A/z + B/(z+2)
Then, we can find the values of A and B by finding a common denominator and equating the numerators:
(z+1)/(z(z+2)) = A/z + B/(z+2)
By solving this equation, we can find that A = 1/2 and B = -1/2. Therefore, f(z) can be written as:
f(z) = (1/2)/z + (-1/2)/(z+2)
Finally, we can integrate this expression counterclockwise over the unit circle. Since the unit circle can be parameterized by z = e^(it), where t goes from 0 to 2π, we can use this parameterization to perform the integration.