# How to evaluate this integral to get pi^2/6:

• hb1547
In summary, the conversation touches upon the value of the integral \int_0^\infty \frac{u}{e^u - 1}, which is known to be \frac{\pi^2}{6}. The question arises if this result can be directly evaluated or if it is obtained through other methods, such as using the equation \zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du. The conversation ends with a mention of using complex variable techniques to solve the equation.
hb1547
$\int_0^\infty \frac{u}{e^u - 1}$

I know that this integral is $\frac{\pi^2}{6}$, just from having seen it before, but I'm not really sure if I can evaluate it directly to show that.

I know that:

$\zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du$

Does the value $\frac{\pi^2}{6}$ come from using other methods of showing the result for $\zeta(2)$ and solving the equation, or is that integral another way of evaluating $\zeta(2)$?

hb1547 said:
$\int_0^\infty \frac{u}{e^u - 1}$

I know that this integral is $\frac{\pi^2}{6}$, just from having seen it before, but I'm not really sure if I can evaluate it directly to show that.

I know that:

$\zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du$

Does the value $\frac{\pi^2}{6}$ come from using other methods of showing the result for $\zeta(2)$ and solving the equation, or is that integral another way of evaluating $\zeta(2)$?

never mind ... my complex variable technique is rusty ...

Last edited:
Anyone else have any input?

## 1. What is the integral that results in pi^2/6?

The integral in question is the famous Basel problem, which can be written as ∫01 (1/x^2) dx.

## 2. How do you solve the integral for pi^2/6?

The integral can be solved using various mathematical techniques, such as the Euler-Maclaurin summation formula or the Riemann zeta function. The resulting value is pi^2/6 or approximately 1.644934.

## 3. Why is pi^2/6 important?

The value of pi^2/6 is significant because it is closely related to the famous mathematical constant pi. It also has various applications in mathematics, physics, and engineering.

## 4. Can the integral be evaluated using a calculator?

No, the integral cannot be evaluated using a calculator as it requires advanced mathematical techniques. However, there are online calculators available that can solve the integral for you.

## 5. What is the significance of the integral in mathematics?

The integral is significant in mathematics as it is a prime example of a convergent series. It also has connections to other important mathematical concepts, such as the Riemann zeta function and the harmonic series.

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