Enraging Pythagoras 2.0

Calculus Level 2

True or False?

If \[ \int \frac{e^x}{1+x^2} \ dx = f(x)e^x + C,\] where \(C\) is a constant, then \(f\) is a rational function. That is, \(f(x) = \frac{\mathrm{P}(x)}{\mathrm{Q}(x)},\) where \(\mathrm{P}(x)\) and \(\mathrm{Q}(x)\) are real polynomials in \(x.\)


Note: This problem was adapted from a question in the 2016 STEP III exam.

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