I'd like to mention that one may prove the irrationality of $\pi$, $\ln 2$, $\zeta(2)$, and $\zeta(3)$ in a relatively uniform way using simple integral representations.

Assume that one wants to show the irrationality of a number $\xi$ which can be presented for every $k\in\mathbb N$ in terms of the moments of some function $f$
$$a_k+b_k\xi=\int_{0}^{1}x^k f(x) dx, $$
where $a_k$, $b_k\in\mathbb Q$. If $\xi$ were rational than the equality might be rewritten as
$$\frac{c_n}{d_n}=\int_{0}^{1}P_n(x)f(x)dx,\quad P_n(x)=\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n),\ n\in\mathbb N,$$
where and $c_n$, $d_n$ are integers. (The choice of the Legendre polynomials $P_n$ allows to perform integrations by parts easily.) Now, if we can show that

$$d_n\left|\int_{0}^{1}\frac{1}{n!}x^n(1-x)^n\frac{d^n}{dx^n}f(x)dx\right|\to 0,$$
this would imply that $c_n\to 0$ which is impossible; so $\xi$ cannot be rational.

The difficult part, of course, is to find the suitable function $f$.

For $\xi=\pi$ we may take $f(x)=\sin\pi x$ and use the fact that $\int_{0}^{1}x^k\sin(\pi x) dx$ is a polynomial in $\pi$ of degree $k$ divided by $\pi^k$. Assuming $\pi=a/b$ we will get that
$$0<|c_n|=\left|a^n\int_{0}^1P_n(x)\sin(\pi x) dx\right|\to 0.$$

For $\xi=\ln 2$ take $f(x)=1/(1+x)$. If $\ln 2$ were $a/b$, then
$$0<|c_n|=\left|bD_n\int_{0}^{1}\frac{P_n(x)}{1+x}dx\right|\to 0$$
(where $D_n={\rm LCM}\{1,2,\dots,n\}$).

For $\xi=\zeta(2)$ the choice is
$$f(x)=\int_{0}^{1}\frac{(1-y)^n}{1-xy}dy,$$
and the assumption $\zeta(2)=a/b$ leads to
$$0<|c_n|=\left|D_{n+1}^2\int_{0}^{1}P_n(x)f(x)dx\right|\to 0$$
(where $D_n={\rm LCM}\{1,2,\dots,n\}$).

Finally, for $\xi=\zeta(3)$ take
$$f(x)=\int_{0}^{1}\frac{P_n(y)}{1-xy}\ln xy\ dy.$$
If $\zeta(3)=a/b$, then
$$0<|c_n|=\left|D_{n+1}^3\int_{0}^{1}P_n(x)f(x)dx\right|\to 0.$$

The irrationality proofs are contained in the book by J.M. Borwein and P.B. Borwein. There is also a nice summary in the note by D. Huylebrouck (with all four proofs occupying less than five pages).