I have an exercise, whose aim is to show that $\pi^2$ is irrational by contradiction. We suppose that $\frac{a}{b} = \pi ^2$ with $a,b \in \mathbb{N} ^*$. We put $$N_n = \frac{\pi a^n}{n!} \int ^{1}_{0}x^n(1-x)^n \sin (\pi x) dx$$
I was first asked to show that $N_n > 0 $ and that $\lim_{n\to\infty} N_n = 0$, which I did.
Then I am asked to show that $\forall n \in \mathbb{N}, N_n \in \mathbb{N}$ and then conclude. I am asked to use integration by parts.
By integrating twice, I find that $N_n$ is also equal to $$ \frac{\pi a^n}{(n-1)!} \int ^{1}_{0}x^n (1-x)^n \sin(\pi x) dx $$ which allows me to conclude that $$\frac{\pi a^n}{n!} = \frac{\pi a^n}{(n-1)!} $$ which is absurd. I guess at this point I managed to prove that $\pi ^2 $ is irrational, but I still want to know how could I proceed to show that $\forall n \mathbb{N}, N_n \in \mathbb{N} $?
You missed some crucial details (or perhaps your book author is trying to be smart by missing out on these and expecting you to generate it yourself). The same exercise is given in Apostol's Mathematical Analysis (problem 7.33 page 180) in a much better fashion and describes Ivan Niven's proof of irrationality of $\pi^{2}$. I add some missing details.
Let $\pi^{2}=a/b$ and $f(x) =x^{n} (1-x)^{n}/n!$ and $$F(x) =b^{n} \sum_{k=0}^{n}(-1)^{k}f^{(2k)}(x)\pi^{2n-2k}$$ The crucial thing to note here is that $f^{(k)} (0)$ and hence $f^{(k)} (1)$ are integers (prove this!) so that $F(0),F(1)$ are integers. Further there is this easily verifiable identity $$\frac{d} {dx} \{F'(x) \sin\pi x-\pi F(x) \cos \pi x\} =\pi^{2}a^{n}f(x) \sin\pi x$$ which upon integration leads to $$F(0)+F(1)=\pi a^{n} \int_{0}^{1}f(x)\sin\pi x\, dx$$ and thus your $N_{n} $ is a positive integer as expected.
Niven's proof is non-obvious and is based on the key ideas first used in Hermite's proof of transcendentality of $e$. It is reasonable to assume that unless one is aware of it, the proof is difficult to come all by itself.
This particular problem was the reason I bought Apostol's book and it turned out to be one of the best books I have. This happened long time ago when there was no Internet access available for me and I was desperate enough to find a proof of irrationality of $\pi$.