Integral Challenge: Evaluating $\int_0^\infty \frac{x^2+2}{x^6+1} \, dx$

In summary, the purpose of evaluating this integral is to find the exact value of the area under the curve of the function $\frac{x^2+2}{x^6+1}$ from 0 to infinity, providing important information about its behavior and properties. It is considered challenging to solve because it involves a rational function with a high degree polynomial, making it suitable for techniques such as partial fraction decomposition or contour integration. Basic integration techniques cannot be used, so specialized methods like trigonometric substitutions are needed. Evaluating this integral is important in mathematics because it allows for a deeper understanding of the function and has practical applications in various fields. It can also be a fun and rewarding exercise for mathematicians and students.
  • #1
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Evaluate $\displaystyle\int\limits_0^{\infty} \dfrac{x^2+2}{x^6+1} \, dx$.
 
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  • #2
We have $$I= \int_0^\infty \frac{x^2+2}{x^6+1}\,\mathrm{dx}$$
Let $x \mapsto \frac{1}{x}$ then$$I= \int_0^\infty \frac{x^2+2x^4}{x^6+1}\,\mathrm{dx}$$
So that $$2I = 2\int_0^\infty \frac{1+x^2+x^4}{1+x^6}\,\mathrm{dx}$$

i.e. $$I = \int_0^\infty \frac{1+x^2+x^4}{1+x^6}\,\mathrm{dx}$$By partial fractions $$\begin{aligned} \frac{1+x^2+x^4}{1+x^6} &= \frac{1}{3}\cdot \frac{1}{1+x^2}+\frac{2}{3}\cdot \frac{x^2+1}{x^4-x^2+1} \\& = \frac{1}{3}\cdot \frac{1}{1+x^2} +\frac{2}{3}\cdot \frac{1+1/x^2}{x^2+\frac{1}{x^2}+1} \\& = \frac{1}{3}\cdot \frac{1}{1+x^2} +\frac{2}{3}\cdot \frac{1+1/x^2}{(x-1/x)^2+1}\end{aligned} $$
This along with the sub $u = x-\frac{1}{x}$ helps us finish off the integral,
$$\begin{aligned} I & = \frac{1}{3}\cdot \int_0^\infty \frac{1}{1+x^2}\, \mathrm{dx} +\frac{2}{3}\cdot \int_0^\infty\frac{1+1/x^2}{(x-1/x)^2+1}\,\mathrm{dx} \\& = \frac{1}{3}\cdot \frac{\pi}{2}+\frac{2}{3} \cdot \int_{-\infty}^{\infty} \frac{1}{u^2+1}\,\mathrm{du} \\& = \frac{\pi}{6}+\frac{2\pi}{3} \\& = \frac{5 \pi}{6}.\end{aligned} $$Therefore $$\int_0^\infty \frac{x^2+2}{x^6+1}\,\mathrm{dx}= \frac{5\pi}{6}.$$
 
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  • #3
Consider the function $\displaystyle f(z) = \frac{z^2+2}{z^6+1}$; $f$ has simple poles at $z = \pm i, \pm i^{\frac{1}{3}}, \pm i^{\frac{5}{3}}$, three of which lie in the upper half plane;
those are $z_1 = i, ~ z_2 = i^{\frac{1}{3}}, ~ z_3 = i^{\frac{5}{3}}$. The sum of the residues at these poles is given by

$\displaystyle \begin{aligned} \sum_{1 \le i \le 3} \text{res} f(z_i) & = \frac{1}{6} (-2 i - \sqrt{3})+\frac{1}{6}(-2 i +\sqrt{3})-\frac{i}{6} \\& = -\frac{5i}{6}\end{aligned} $​

Now, let $\Gamma$ be the upper half-plane semi-circle $|z| = a$ with radius $a$ in counterclockwise direction.

Since $z_1, z_2, z_3$ all lie in the upper half plane, by the residue theorem we have

$\displaystyle \begin{aligned} \oint_\Gamma f(z)\,dz & =2i \pi \sum_{1 \le i \le 3} \text{res} f(z_i) \\& = 2i \pi \cdot \frac{-5i}{6} \\& = \frac{5\pi}{3} \end{aligned}$​
But also
$\displaystyle \oint_\Gamma f(z)\,dz=\int_{-a}^{a} f(z) \,\mathrm{dz} +\int_{\text{Arc}} f(z)\,\mathrm{dz}$​

Therefore

$\displaystyle \int_{-a}^{a} f(z) \,\mathrm{dz} +\int_{\text{Arc}} f(z)\,\mathrm{dz} = \frac{5\pi}{3}$​

By taking $a \to \infty$, since

$ \displaystyle \bigg| \int_{\text{Arc}} f(z) \,\mathrm{dz} \bigg| \le \frac{a^2+2}{a^6-1}~ a \pi \longrightarrow 0$​
we have $\displaystyle \int_{\text{Arc}} f(z)\,\mathrm{dz} \to 0$ as $a \to \infty$, therefore

$\displaystyle \int_{-\infty}^{\infty}\frac{z^2+2}{z^6+1}\,\mathrm{dz} = \frac{5\pi}{3} $​

Therefore (since $f$ is an even function),

$\displaystyle \int_{0}^{\infty}\frac{z^2+2}{z^6+1}\,\mathrm{dz} = \frac{5\pi}{6}. $​
 
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FAQ: Integral Challenge: Evaluating $\int_0^\infty \frac{x^2+2}{x^6+1} \, dx$

1. What is the purpose of the Integral Challenge?

The purpose of the Integral Challenge is to evaluate the integral 0 (x²+2)/(x⁶+1) dx and showcase the process of solving a challenging integral problem.

2. What techniques can be used to solve this integral?

There are several techniques that can be used to solve this integral, including substitution, partial fractions, and contour integration. Each technique has its own advantages and may be more suitable depending on the specific problem at hand.

3. Is there a specific approach that should be followed when solving this integral?

There is no one specific approach that should be followed when solving this integral. It is important to carefully analyze the integrand and choose a technique that best fits the problem. It may also be helpful to break the integral into smaller parts and apply different techniques to each part.

4. How can I check if my solution to the integral is correct?

One way to check the correctness of your solution is to use a computer algebra system or integral calculator to evaluate the integral. You can also check your solution by taking the derivative of your answer and verifying that it matches the original integrand.

5. Are there any real-world applications of this integral?

Yes, this integral can be used to calculate the expected value of a continuous random variable with a probability density function of (x²+2)/(x⁶+1). It can also be applied in engineering and physics problems involving power functions and harmonic oscillators.

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