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Determine $G(x)$.$$G(x)=\int_{0}^{x} \sqrt{1-t^{2}} d t$$

$$\sqrt{1-x^{2}}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 7

Substitution and Properties of Definite Integrals

Integrals

Campbell University

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.

01:56

Determine $G(x)$.$$G(x…

02:15

01:20

Find $G^{\prime}(x)$$$…

01:43

02:46

01:57

01:53

01:15

Find $G^{\prime}(1),$ wher…

01:22

00:51

Find $G^{\prime}(x)$.$…

02:05

Please answer ASAP.

Okay, So our task in this problem is to find the derivative of G, and they give us The original function is defined as the integral from zero to X of the square root of one minus t squared D t. So, uh, the theme in this problem is the fundamental theorem of capitalists. And here's the premises. If I don't know the anti derivative of this, it's, uh too complicated for calculus one. But if you were to simplify this by doing the anti derivative, you would have some function from zero to X, and you would plug in those X values wherever, wherever they go in for tea, and then you would subtract off. So some function with accident, I'll call it like little G of X minus, um, some constant, because the constants representing plugging zero in for those values. But then, if I were to ask you for the derivative of this So, uh, let me do capital g prime of X. But what you do is then the derivative of each piece and what's the derivative of a constant zero. So what? What the fundamental theorem of calculus basically says is that we can just replace the anti terror of the integral, and the derivative will cancel each other out, and you can be left with just plugging this in. Now. There's a little bit more to the fundamental theorem of calculus because it's essentially the chain role. Well, what's nice about this problem is, if you were to do the derivative of X, it would just be times one, um, so we don't have to write that. That's why this problem is done right here.

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