To exculpate someone is to prove that they are not guilty of doing something wrong.

// The editorial expresses confidence that the evidence will exculpate the accused.

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“Research shows that social-media use is associated with greater narcissism (as well as depression and anxiety).... But it is too easy to exculpate ourselves as a society by pointing to technology and trends we can scarcely control, and young adults may not be in a position to address their avoidant behavior. The rest of us can help.” — Arthur C. Brooks, The Atlantic, 1 June 2023

There’s no need to say “my bad” if you’re unfamiliar with exculpate; while the word is far from rare, it is most often encountered in formal writing in reference to the clearing of someone of alleged fault or guilt, as in “they were exculpated of any wrongdoing.” You may be more familiar with a pair of terms that, like exculpate, come from the Latin noun culpa, meaning “blame” or “guilt.” One is the adjective culpable, used to describe someone deserving of condemnation or blame. The other is the Latin phrase mea culpa, which translates directly as “through my fault” and refers to an acknowledgement of personal fault or error that is more formal than, well, “my bad.”