If the integral of ((x² + 1)*e^x) / (x + 1)² with respect to x equals f(x)*e^x + C (where C is an arbitrary constant), then the value of the third derivative of f with respect to x evaluated at x = 0 is:

Question

Accepted Answer

12. Matching f(x)+f'(x) = (x²+1)/(x+1)² gives f(x) = (x-1)/(x+1) = 1 - 2/(x+1); differentiating three times yields f'''(x) = 12/(x+1)⁴, so f'''(0) = 12.

If the integral of ((x² + 1)e^x) / (x + 1)² with respect to x equals f(x)e^x + C (where C is an arbitrary constant), then the value of the third derivative of f with respect to x evaluated at x = 0 is:

Solution

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