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Find $f(x)$.$$f^{\prime \prime}(x)=2, f^{\prime}(1)=2, f(1)=3$$

$$x^{2}+2$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Campbell University

Baylor University

Idaho State University

Boston College

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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All right. We want to find F of X. Given that F double prime of X equals two. F prime one equals 2.5 of one equals three. This question is asking us to solve the second order differential equation specifically, it's asking this is all an initial value problem where we're giving the initial conditions for the derivative and for the function since F double prime of X equals two is solely a function of X. It doesn't involve any other variables. We can simply solve this problem by taking anti derivative successively. So the first anti derivative is two. X plus C. Constant. Immigration plugging in. F prime one equals two, gives us two equals two plus C or C equals zero. Thus F F prime access to X. And now we take another anti derivative. So now we have F prime active X equals X squared plus D. The second class of integration, solving for X one equals three gives us three equals one plus D, or D equals two. Thus we saw for our final function out of X. And we have F of X is equal to X squared plus the constant, too

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