We can actually deduce here that Gage changed such an integral part of Truth's speech because she wanted to create an impression that the black women experienced helplessness in their fight for equality. Also, it actually reveals that Gage wanted Truth to sound like a Southern slave.
An integral part of something actually refers to the main and important part of that thing. Integral part usually tells how important something is.
We see here that Gage changing some integral part of Truth's speech shows she was actually distorting the original to add hers.
Gage Frances Dana was known to be an abolitionist, reformer and an activist.
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