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Find $f(x)$.$$f^{\prime}(x)=3-2 / x+x^{2}, f(1)=5$$

$$3 x-2 \ln |x|+\frac{x^{3}}{3}+\frac{5}{3}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Harvey Mudd College

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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Find $f(x)$.$$f^{\prim…

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find $f^{\prime \prime}(2)…

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we want to find ffx given that F prime of X is equal to three minus two X plus x square. For initial conditions, afford one equals five. This question is challenging us to solve a differential equation for Activex specifically, it's asking us to solve an iVP and initial value problem since F prime of X is entirely a function of X itself. Rather than using a different differential equation method of solving, we're simply going to take the anti derivative effect crime and then use the initial conditions to solve for F of X explicitly. So let's first take the anti derivative turn. By turn, we contain a five X equals three X minus two. Natural logarithms. X plus X cubed over three plus the constant of integration. See next let's use F. One equals five to solve for C. So five equals three plus one third plus C. Because natural organism wanna zero. Thus C. Is equal to five thirds. Now we can plug into our F. To have our final solution. This gives ethical three x minus two. Natural organism. X plus X cubed over three plus the constant 5/3.

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