Problem: Definite Integral with R(Polynomials): int[0,1](x^2+1)/(x^4+1)dx

[Vect]

\(\displaystyle \displaystyle \int_0^1\, \)\(\displaystyle \dfrac{x^2\, +\, 1}{x^4\, +\, 1}\, dx\)

I suppose it could be solved using partial fraction decomposition, but I'm sure there must be a trick to this. My upcoming exam requires me to be very quick, so I hope that justifies the question's nature. Anyone have any ideas? Perhaps a clever notation?

 

[pka]

\(\displaystyle \displaystyle \int_0^1\, \)\(\displaystyle \dfrac{x^2\, +\, 1}{x^4\, +\, 1}\, dx\)

I suppose it could be solved using partial fraction decomposition, but I'm sure there must be a trick to this. My upcoming exam requires me to be very quick, so I hope that justifies the question's nature. Anyone have any ideas? Perhaps a clever notation?

I agree with you that this would be an unreasonable question to put on an ordinary exam. Partial fraction is the way to go with this, unless you are up to speed with series.

A good test maker would use something like this: \(\displaystyle \displaystyle\int_0^\pi {\frac{{{x}}}{{{x^4} + 1}}dx} \).

That question is straightforward and accomplishes the same objective.