∫ 3x² + 1 / (x² - 1)³ dx equals (where K is constant of integration)
Step-by-Step Solution
Key Concept: The integral can be solved by rewriting the numerator as (x^2 - 1) + 2(x^2 + 1) or by observing the derivative of x/(x^2-1)^2. Alternatively, divide numerator and denominator by x^6 to simplify.
Let I = \int (3x^2 + 1) / (x^2 - 1)^3 dx. Divide numerator and denominator by x^6: I = \int (3/x^4 + 1/x^6) / (1 - 1/x^2)^3 dx. Let 1 - 1/x^2 = t, then (2/x^3) dx = dt. The integral becomes \int (3/x^4 + 1/x^6) / t^3 dx. This can be simplified to \int (1/x^2(3/x^2 + 1/x^4)) / t^3 dx. A simpler approach is to check the derivative of the options. Let f(x) = -x / (x^2 - 1)^2. f'(x) = - [ (x^2 - 1)^2 - x * 2(x^2 - 1) * 2x ] / (x^2 - 1)^4 = - [ (x^2 - 1) - 4x^2 ] / (x^2 - 1)^3 = - [ -3x^2 - 1 ] / (x^2 - 1)^3 = (3x^2 + 1) / (x^2 - 1)^3. Thus, the integral is -x / (x^2 - 1)^2 + K.
Correct Answer: 3