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Find $f(x)$.$$f^{\prime}(x)=3 e^{x}-\frac{2}{x^{2}}+1, f(2)=7$$

$$3 e^{x}+\frac{2}{x}+x+4-3 e^{2}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

02:17

Find $f^{\prime \prime}(x)…

00:22

Find $f^{\prime}(x)$.$…

01:35

Find $f^{\prime}(x)$$$…

01:23

00:32

Uh huh. We want to find F of X. For F prime of X equals three E. To the x minus two X squared plus one. We're F of two equal seven. This question is challenging us to solve a differential equation for F specifically, it's asking you to solve the initial value problem given instead of using a standard method for solving differential equations. Since F prime of X is simply in terms of X, we can take the anti derivative of F. Prime to solve for F. And then using the initial conditions can find the explicit solution. So let's take the anti derivative F is three E. D X plus two over X plus X plus a constant of integration. See next week is our initial conditions to solve for C. So we have seven equals three E squared plus one plus two plus C solving give C equals four minus three squared. This is from F two equals seven. Thus we have final solution F of X is equal to three E. D x plus two over X plus X plus four minus three E squared.

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