Catalan's Constant

By Brad Rodgers on Tuesday, December 18, 2001 - 09:09 pm:

How do you deduce that

$\int_{0}^{π / 2} x / \sin (x) dx = 2 G$ ?

for $G = \sum_{n = 0}^{\infty} (- 1)^{n} / (2 n + 1)^{2}$ .
Brad

By Yatir Halevi on Tuesday, December 18, 2001 - 09:27 pm:

I guess you already know this, but anyway:
The integral of x/sin(x) is:
x+x³ /18+7x⁵ /1800+...+2(2^2n-1 -1)b_n x²ⁿ⁺¹ /(2n+1)!+...
Where b_n is the nth Bernoulli number.

Yatir

By Michael Doré on Wednesday, December 19, 2001 - 01:57 am:

I'll let Yatir get back to you on that formula, but to solve your original question you can use:

$1 - t^{2} + t^{4} - \dots = 1 / (1 + t^{2})$

which is an identity for $t$ in $(- 1, 1)$ .

Integrating from 0 to $x$ we obtain:

$x - x^{3} / 3 + x^{5} / 5 - \dots = \arctan x$

Dividing through by $x$ and integrating from 0 to 1 gives:

$1 / 1^{1} - 1 / 3^{2} + 1 / 5^{2} - \dots = \int_{0}^{1} (\arctan x) / x dx$

This gives the result because the integral on the right hand side is actually half your integral (to see this substitute $u = 2 \arctan x$ ).