Definite Integration
Indefinite Integration
Grade Class 12
Question:
The integral ∫ \frac{3x^2 + 1}{(x^2 - 1)^3} dx equals (where K is constant of integration)
\frac{x}{(x^2 - 1)^2} + K
K - \frac{x}{(x^2 + 1)^2}
K - \frac{x}{(x^2 - 1)^2}
K - \frac{x^2}{(x + 1)^2}
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.
Let I = \int (3x^2 + 1) / (x^2 - 1)^3 dx. Notice that d/dx [x / (x^2 - 1)^2] = [(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. Thus, \int (3x^2 + 1) / (x^2 - 1)^3 dx = - \int d/dx [x / (x^2 - 1)^2] dx = - x / (x^2 - 1)^2 + K.
Correct Answer: C