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Numerade Educator

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Problem 3 Easy Difficulty

Verify by differentiation that the formula is correct.

$ \displaystyle \int \tan^2 x \,dx = \tan x - x + C $

Answer

$\tan ^{2} x+1-1$

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Video Transcript

right, we can prove that that was the correct anti derivative. If we take the derivative of the right side and um X minus x plus C. Um and I'm not going to work through the proof as to why the derivative of tangent is seeking squared of X. Uh And then the derivative of X is one and then the derivative of a constant zero. And my entire goal is to get this piece, I just found to be equal to the left side of the equation, which was tangent squared effects. Um Now I would expect my students to know directly from the step to this step. So it is correct. Other people might be familiar with the trig identity that tangent squared of X plus one is equal to seek and squared. Well, you can easily fix that by subtracting one over or maybe what I could do is replace this second squared With tangent squared effects plus one. And then subtract this one off to cancel them out and your left here. So really I need to show all of these steps um to get from here to here, but that's what they wanted you to do is the steps to get to the left side, starting with the right. Mhm.