Evaluation of $\int_0^\infty \frac{x^2}{1+x^5} \mathrm{d} x$ by contour integration

Question

Consider the following integral: $$\int_{0}^{\infty} \frac{x^{2}}{1+x^{5}} \mathrm{d} x \>.$$

I did the following: Since $-1$ is a pole on the real axis, I took $z_{1}=e^{3\pi/5}$ then constructed an arc between $Rz_{1}$ and $R$ :

$f(e^{2\pi i /5}z)=f(z)$ it follows that : $(1-e^{2pi i/5})\int_{\alpha}f(z)dz + \int_{\beta}f(z)dz = 2\pi i \text{ Res } z_{2} f $ ,
where $\alpha$ is the way along the eral axis, $\beta$ is the way along the arc and $z_{2}=e^{i\pi/5}$

This gives: $\int_{0}^{\infty}...dx = 2\pi i (\frac{e^{2i\pi/5}}{-5})\cdot \frac{1}{(1-e^{2\pi i/5})}$

The real part of the right side in wolframalpha does not give the same result as the left side.

Does anybody see why? Please do tell.

Answer 1

I think you want to use the curve $e^{2\pi i/5}x$ for the inbound curve circling the pole at $e^{\pi i/5}$, which has residue $\frac1{5e^{2\pi i/5}}$.

Because the integral along $|z|=R$ vanishes as $R\to\infty$, we get $$ \left(1-e^{6\pi i/5}\right)\int_{0}^{\infty} \frac{x^{2}}{1+x^{5}}\;\mathrm{d}x=2\pi i\frac{1}{5e^{2\pi i/5}} $$ Which gives $$ \int_{0}^{\infty} \frac{x^{2}}{1+x^{5}}\;\mathrm{d}x=\frac{2\pi i}{5\left(e^{2\pi i/5}-e^{-2\pi i/5}\right)}=\frac{\pi}{5\sin(2\pi/5)} $$

Answer 2

I ran out of patience, but the partial fraction approach can give an indefinite integral in closed form, as the tenth roots of 1 have real and imaginary parts that can be written with nothing worse than square root signs. I got $$ \frac{x^2}{1 + x^5} \; = \; \; \frac{A}{x + 1} + \frac{B_1 x + B_0}{x^2 - \frac{1}{2}(\sqrt 5 + 1)x + 1} + \frac{C_1 x + C_0}{x^2 + \frac{1}{2}(\sqrt 5 - 1)x + 1}.$$ Then I got $$ A = \frac{1}{3}, \; B_0 = B_1 = \frac{1}{6}(\sqrt 5 - 1), \; \; C_0 = C_1 = -\frac{1}{6}(\sqrt 5 + 1).$$ The two quadratic denominators terms are positive over the reals. The rest of the task is also cookbook, you rewrite, for example $B_1 x + B_0$ as $\frac{B_1}{2}$ times the derivative of $x^2 - \frac{1}{2}(\sqrt 5 + 1)x + 1$ plus a constant term , call it $B_3.$ So, for this part, you get the indefinite integral being $$ \frac{B_1}{2} \log(x^2 - \frac{1}{2}(\sqrt 5 + 1)x + 1) + B_3 \arctan(linear) $$ I imagine you can force your computer algebra system to show all the steps in this calculation while getting the specific numbers right. The beginning is to define $$ \omega = e^{\pi i / 5} = \frac{1}{4}(1 + \sqrt 5) + \frac{i}{4} \sqrt{10 - 2 \sqrt 5} $$ with $$ 1 + x^5 = (x + 1)(x - \omega)(x - \omega^3)(x - \omega^7)(x - \omega^9) $$