How to align two or more groups of equations across \items in an enumerated environment?

Question

In the snip, we can see that two groups of equations, aligned individually on the within-group = symbols.

\documentclass{article}
\usepackage{mathtools}  
\begin{document}  
\begin{enumerate}
\item
    $$\begin{aligned}
    \int x^2 \cos x \,dx
    &= x^2 \sin x - 2 \int x \sin x \,dx
    \\ &= x^2 \sin x - 2 (-x \cos x +\int \cos x \,dx)
    \\ &= x^2 \sin x - 2 (-x \cos x + \sin x \,dx + C)
    \\ &= x^2 \sin x + 2x \cos x - 2\sin x + C
    \end{aligned}$$

\item
    $$\begin{aligned}
    \int \frac{\ln x}{x^3} \,dx
    &= -\frac{1}{2} \frac{\ln x}{x^2} +
        \frac{1}{2} \int \frac{1}{x^3} \,dx
    \\ &= -\frac{\ln x}{2x^2} +
        \frac{1}{2} \cdot -\frac{1}{2x^2} + C
    \\ &=  -\frac{\ln x}{2x^2} - \frac{1}{4x^2} + C
    \\ &= -\frac{1}{4x^2}(2\ln x + 1) + C
    \end{aligned}$$ 
\end{enumerate}  
\end{document}

How to align the two groups of equations on the same equal symbol "="?

Like this:

\documentclass{article}  
\usepackage{mathtools}
\begin{document}
\begin{enumerate}
\item
    $$\begin{aligned}
    \int x^2 \cos x \,dx
    &= x^2 \sin x - 2 \int x \sin x \,dx
    \\ &= x^2 \sin x - 2 (-x \cos x +\int \cos x \,dx)
    \\ &= x^2 \sin x - 2 (-x \cos x + \sin x \,dx + C)
    \\ &= x^2 \sin x + 2x \cos x - 2\sin x + C
    \\
    \\
    \int \frac{\ln x}{x^3} \,dx
    &= -\frac{1}{2} \frac{\ln x}{x^2} +
        \frac{1}{2} \int \frac{1}{x^3} \,dx
    \\ &= -\frac{\ln x}{2x^2} +
        \frac{1}{2} \cdot -\frac{1}{2x^2} + C
    \\ &=  -\frac{\ln x}{2x^2} - \frac{1}{4x^2} + C
    \\ &= -\frac{1}{4x^2}(2\ln x + 1) + C
    \end{aligned}$$
\end{enumerate}
\end{document}

In this approach, the numbers generated by enumerate are missing, so I'm wondering are there better ways.

Answer 1

I suggest you (a) use a single align* environment across all items and (b) use \intertext instructions in which you execute \item \phantom{x}. (The \phantom{x} directive -- or something similarly invisible -- helps make LaTeX believe that there's actually some material associated with the \item.)

\documentclass{article}
\usepackage{mathtools}
\allowdisplaybreaks
\begin{document}

\begin{enumerate}
\item
\begin{align*}
    \int x^2 \cos x \,dx
    &= x^2 \sin x - 2 \int x \sin x \,dx \\
    &= x^2 \sin x - 2 (-x \cos x +\int \cos x \,dx) \\
    &= x^2 \sin x - 2 (-x \cos x + \sin x \,dx + C) \\
    &= x^2 \sin x + 2x \cos x - 2\sin x + C \\
\intertext{\refstepcounter{enumi}\labelenumi}
    \int \frac{\ln x}{x^3} \,dx
    &= -\frac{1}{2} \frac{\ln x}{x^2} +
        \frac{1}{2} \int \frac{1}{x^3} \,dx \\
    &= -\frac{\ln x}{2x^2} +
        \frac{1}{2} \cdot -\frac{1}{2x^2} + C \\
    &=  -\frac{\ln x}{2x^2} - \frac{1}{4x^2} + C \\
    &= -\frac{1}{4x^2}(2\ln x + 1) + C
\end{align*}
\end{enumerate}

\end{document}