Question

The value of the integral $\int_{0}^{\infty} \frac{x-1}{x^8-1} dx$ is

Answer 1

$\frac{\pi\sqrt{2}}{8}$

Answer 2

The integral to evaluate is $I = \int_{0}^{\infty} \frac{x-1}{x^8-1} dx$.

1. Simplify the integrand: The denominator can be factored as $x^8-1 = (x-1)(x^7+x^6+x^5+x^4+x^3+x^2+x+1)$. For $x \neq 1$, we can cancel $(x-1)$ from the numerator and denominator: $\frac{x-1}{x^8-1} = \frac{1}{x^7+x^6+x^5+x^4+x^3+x^2+x+1}$. The singularity at $x=1$ is removable, so the integral can be evaluated using the simplified form.

2. Apply a known integral identity: The integral is of the form $\int_{0}^{\infty} \frac{x^a - x^b}{x^n - 1} dx$. For this type of integral, when the singularity at $x=1$ is removable (i.e., $x^a-x^b$ has a root at $x=1$), the value is given by the formula: $\int_{0}^{\infty} \frac{x^a - x^b}{x^n - 1} dx = \frac{\pi}{n} \left( \cot\left(\frac{(b+1)\pi}{n}\right) - \cot\left(\frac{(a+1)\pi}{n}\right) \right)$, provided that $-1 < a, b < n-1$.

3. Identify parameters: In our integral, $\frac{x-1}{x^8-1}$: $a = 1$ (power of $x$ in the first term of the numerator) $b = 0$ (power of $x$ in the second term of the numerator, as $1 = x^0$) $n = 8$ (power in the denominator). All conditions are met: $-1 < 0, 1 < 7$.

4. Substitute values into the formula: $I = \frac{\pi}{8} \left( \cot\left(\frac{(0+1)\pi}{8}\right) - \cot\left(\frac{(1+1)\pi}{8}\right) \right)$ $I = \frac{\pi}{8} \left( \cot\left(\frac{\pi}{8}\right) - \cot\left(\frac{2\pi}{8}\right) \right)$ $I = \frac{\pi}{8} \left( \cot\left(\frac{\pi}{8}\right) - \cot\left(\frac{\pi}{4}\right) \right)$.

5. Evaluate cotangent terms: We know $\cot\left(\frac{\pi}{4}\right) = 1$. For $\cot\left(\frac{\pi}{8}\right)$, use the half-angle identity $\cot(\theta) = \frac{1+\cos(2\theta)}{\sin(2\theta)}$. Let $2\theta = \frac{\pi}{4}$, so $\theta = \frac{\pi}{8}$. $\cot\left(\frac{\pi}{8}\right) = \frac{1+\cos(\frac{\pi}{4})}{\sin(\frac{\pi}{4})} = \frac{1+\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = \frac{2+\sqrt{2}}{\sqrt{2}} = \sqrt{2}+1$.

6. Calculate the final value: $I = \frac{\pi}{8} ((\sqrt{2}+1) - 1)$ $I = \frac{\pi}{8} (\sqrt{2})$ $I = \frac{\pi\sqrt{2}}{8}$.

The final answer is $\frac{\pi\sqrt{2}}{8}$.