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Let $ \displaystyle F(x) = \int^x_2 e^{t^2} \, dt $. Find an equation of the tangent line to the curve $ y = F(x) $ at the point with $ x $-coordinate 2.

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01:15

Frank Lin

Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Integration

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

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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Okay. The first thing we can do is we can differentiate using the fundamental form of calculus. We get heated, the X squared. Now plug in half. Promise to We have each of the two squared, which is either the fourth. So, you know, the slope of the tangent is each of the fourth. Now, half of two is the integral from 2 to 2. E t. Square, D t. You know, this is your right. The tangent process for two comma zero. Therefore, the equation is going to be Why might a syrup is you? The fourth, The slope Times X minus two. This is the point slope form equation that you probably did years ago in algebra. So this is our final solution.

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