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Verify by differentiation that the formula is correct.

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

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00:31

Frank Lin

00:25

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Oregon State University

Baylor University

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Verify by differentiation …

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Verify the formulas by dif…

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Use differentiation to ver…

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Use trigonometric identiti…

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.

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