How to calculate $\int_0^\infty \frac{dx}{1+x^6}$

Question

Whenever I tried to do, it failed. Is there anyone to help? $$\int_0^\infty \frac{dx}{1+x^6}$$

Answer 1

Hint: find the roots of $x^6=−1$ in the complex plane, then by the factor theorem $$x^6+1=(x-x_1)(x-x_2)\cdots(x-x_6)=(x^2+b_1x+c_1)(x^2+b_2x+c_2)(x^2+b_3x+c_3)$$ And apply partial fractions.

Answer 2

Since your integral is even, we can write it as $$ \frac{1}{2}\int_{-\infty}^{\infty}\frac{dz}{z^6 + 1} $$ Let $z = re^{i\theta}$. Then $z^6 = r^6e^{6i\theta} = e^{i(\pi + 2\pi k)}$. So $r = 1$ and $\theta = \frac{\pi}{6} + \frac{\pi k}{3}$. Also, denote $g(z) = z^6 + 1$ so $g'(z) = 6z^5$ which is only zero iff $z = 0$; therefore, $1/g$ only has simple poles.

1. Determine poles in upper half plane.
2. $\frac{1}{2}\int = \pi i\sum_{z_j\in\text{UHP}}\text{Res}_{z_j}\frac{1}{g'(z)}$

This should all be fairly straight forward.

Answer 3

How to calculate $\displaystyle\int_0^\infty\frac{dx}{1+x^6}$ ?

By letting $t=\dfrac1{1+x^6}$ and recognizing the expression of the beta function in the new

integral, then using Euler's reflection formula for the $\Gamma$ function. All integrals of the

form $\displaystyle\int_0^\infty\frac{x^k}{a^n+x^n}~dx~$ can be evaluated in this manner.