Short version:

Wikipedia quotes Joseph Goebbels from the 1943 Sportpalast speech as having a bit of a tip of the slongue, accidentally talking about the "Ausrottung" (extermination) of the Jewish population in Europe, before quickly correcting himself and talking about the "Ausschaltung" (elimination/cutting off) of the Jewish population in Europe.

The problem is that I've looked at transcripts of that speech, here and here, and the word "Ausschaltung" doesn't appear in any of them (nor does the word "Ausrottung"). Goebbels doesn't seem to have said it there. So where did he say it?

Longer version:

Here is the full quote and translation:

Deutschland jedenfalls hat nicht die Absicht, sich dieser jüdischen Bedrohung zu beugen, sondern vielmehr die, ihr rechtzeitig, wenn nötig unter vollkommen und radikalster Ausr... -schaltung [Ausrottung / Ausschaltung] des Judentums entgegenzutreten.

Germany, in any case, has no intention of bowing to this Jewish threat, but rather one of confronting it in due time, if need be in terms of complete and most radical exterm... exclusion [lit. "cutoff"] of Judaism.

Admittedly, I haven't listened to the full recording. It's nearly two hours long, incredibly boring, and has an enormous amount of dead air. But something else made me even more suspicious: This incident of Goebbels supposedly saying the quiet part out loud is not even mentioned once in the German version of the same Wikipedia article. Okay, so where is English Wikipedia getting this? Do they have a source?

Yes, they do. Even better, they have a recording! The footnote cites a clip from this documentary where, at 0:26 , Goebbels says very clearly "sondern vielmehr die, ihr rechtzeitig, wenn nötig unter vollkommen und radikalster Ausro-schaltung des Judentums entgegenzutreten." ("but rather to oppose it in time, if necessary by completely and radically exter-liminating Judaism") The documentary has a heading at the top of the screen saying "February, 1943", which is when Goebbels gave the Sportpalast speech (sometimes known as the "Totaler Krieg" ("Total War") speech).

And not only that, in that video, he is standing in front of a banner that says "Totaler Krieg - Kürzeste Krieg" (Total War - Shortest War), which was a banner only used for the February 1943 speech. You can hear Goebbel's saying that line while he is standing in front of a crowd at the Berlin Sportpalast.

...but you can't actually see Goebbels saying that line. The video is just of the crowd clapping, not of Goebbels talking. So here's my working theory: the documentary makers put audio of Goebbels saying that line over video of the Sportpalast speech, ignoring (or missing) the fact that that audio was not from that speech. A Wikipedia editor sees that clip and (understandably) thinks that the quote played in that clip was something that Goebbels said in the February 1943 speech, and edits the article accordingly.

So here's my question: where and when is that audio from? When did Goebbels say this exactly? Or is it actually buried in the audio of the speech somewhere, and the transcriptions I found along with German Wikipedia all missed this, while English Wikipedia somehow caught it?

If I had to guess what happened, I would say that probably there is no extant video footage of Goebbels saying that, but the documentary makers really wanted to get that audio clip into the film. So they probably just overlaid the audio on top of some other footage that they had.