I have been trying to solve this question and have looked to see if this has been asked but it has not. I am given this assumption:
\begin{equation} \|f(x)\|^2 = 1 \end{equation}
From there I know to rewrite this given expression as an integral using the definition of a norm of a function.
\begin{equation} \|f(x)\|^2 = 1 = \sqrt{\int_{-\infty}^{\infty} |f(x)|^2 \,dx }^2 = \int_{-\infty}^{\infty} |f(x)|^2 \,dx \end{equation}
After simplifying, the part that I do not know how to do is how to solve:
\begin{equation} \int_{-\infty}^{\infty} |f(x)|^2 \,dx \end{equation}
I do not have an expression to replace f(x) with. I do know that I am supposed to solve using integration by parts but I do not know how. Any help is appreciated and sorry is this is confusing!
Your question is rather confusing as it stands. Technically, one could apply integration by parts to this equation by assuming there exists a function $g(x)$ such that $g'(x) = f(x)$. One could then rewrite this, via integration by parts, as $$ \int f^*(x) f(x) \, dx = \int f^*(x) g'(x) \, dx = f^*(x) g(x) - \int g(x) {f^*}'(x) \, dx. $$ (I'm assuming complex-valued functions here. If they're real-valued, just drop the asterisks.) Similarly, you could show that $$ \int f^*(x) f(x) \, dx = x |f(x)|^2 - \int x \left( f^* f' + {f^*}' f \right) \, dx. $$ But without further knowledge about the form of $f(x)$, it is unlikely that these facts will help you. If you've been given the hint to use integration by parts to perform this integral for a specific $f(x)$, it seems more likely to me that there is a different decomposition that allows this to be done.