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If m is a non-zero real number and the integral of x^(5m-1) / (x^(2m) + x^m + 1)³ dx equals f(x) + C, then f(x) is

1. x^(5m) / [2m*(x^(2m) + x^m + 1)²]
2. x^(4m) / [2m*(x^(2m) + x^m + 1)²]
3. x^(5m) / [m*(x^(2m) + x^m + 1)²]
4. x^(4m) / [m*(x^(2m) + x^m + 1)²]

Correct answer: x^(5m) / [2m*(x^(2m) + x^m + 1)²]

Solution

Let t = x^m, dt = m*x^(m-1) dx, so dx = dt/(m*t^(1-1/m)*x^(m-1)). It is cleaner to divide numerator and denominator by x^(3m): numerator becomes x^(5m-1-3m) = x^(2m-1), denominator becomes (1 + x^(-m) + x^(-2m))³. Let u = x^(-2m) + x^(-m) + 1; differentiation gives du = (-2m*x^(-2m-1) - m*x^(-m-1)) dx. The integral reduces to -1/(4m) * (-1/u²) + C = x^(5m)/[2m*(x^(2m)+x^m+1)²] + C after back-substitution.

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