Step-by-Step Solution
Key Concept: The integral can be solved by rewriting the numerator as (x^5+x+1)' - (x^5-4x^4+x+1) or by observing the derivative of the denominator. Alternatively, divide numerator and denominator by x^10 to simplify.
Let I = \int (5x^4 + 4x^5) / (x^5 + x + 1)^2 dx. Divide numerator and denominator by x^10: I = \int (5/x^6 + 4/x^5) / (1 + 1/x^4 + 1/x^5)^2 dx. This is a standard form for rational functions. Alternatively, notice that d/dx (x^5+x+1) = 5x^4+1. The integral is \int (5x^4+1 + 4x^5-1) / (x^5+x+1)^2 dx = \int (5x^4+1)/(x^5+x+1)^2 dx + \int (4x^5-1)/(x^5+x+1)^2 dx. This simplifies to -1/(x^5+x+1) + \int (4x^5-1)/(x^5+x+1)^2 dx. Checking options, (A) and (C) are correct.
Correct Answer: 1, 3