Final answer:
The integral of x/(1 - x^2) from negative infinity to infinity is 0 because the function is odd, and the areas under the curve from negative to zero and zero to positive infinity cancel each other out. Option b is the correct answer.
Step-by-step explanation:
The question asks to evaluate the integral of x/(1 - x^2) from negative infinity to infinity. This integral is called an improper integral because it is taken over an infinite interval. In such cases, the integral may converge (result in a finite number) or diverge (go to infinity). Here, the function, x/(1 - x^2), is an odd function because its graph is symmetric with respect to the origin. When integrated over symmetric limits that include 0, the area under the curve on one side of the y-axis cancels out the area on the other side.
When we break the integral into two parts, from negative infinity to 0 and from 0 to infinity, due to the odd nature of the function, both parts will be equal in magnitude but opposite in sign. This causes them to cancel each other out, leading to a resulting integral value of 0. This result is in line with what we expect from the integral of an odd function over symmetric limits that include 0. Therefore, the correct answer is b) 0.