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Determine $G(x)$.$$G(x)=\int_{1}^{x} t^{4} \sqrt[3]{1-t^{2}} d t$$

$$x^{4} \sqrt[3]{1-x^{2}}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 7

Substitution and Properties of Definite Integrals

Integrals

Oregon State University

University of Michigan - Ann Arbor

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.

01:56

Determine $G(x)$.$$G(x…

01:20

Find $G^{\prime}(x)$$$…

01:57

01:43

01:53

02:46

00:51

Find $G^{\prime}(x)$.$…

01:22

00:43

01:15

Find $G^{\prime}(1),$ wher…

02:05

Please answer ASAP.

02:01

Find the Integral

Yeah. So our task in this problem is to find the derivative. Keep Prime of X. Um When we're giving that G. Of X Is equal to the integral from 0 to X. Oh excuse me can't do zero on this ground. 12 X. of T. to the 4th Times The Cube Root of one minus t square TT. Now here's the thing is a Some students would try to use U. substitution in this problem. This is something you learn in calculon. But if you did the derivative of this negative to t. uh d. T. Um there's no way to get rid of this piece. Now there may be a method of doing this in a help to maybe even help three. Um But a better avenue to do this problem is the fundamental theorem of calculus FTC. And what it basically does is that we need to replace. So the anti guerilla in the derivative cancel each other out. Um and we'll let me put this way, if we did the anti directive it would be from one to action And we would replace the X. M. for all of the teams that we have. We would also replace the one and for all these all the teas. Well when you replace all of the ones and for tease you get some constant. Well what's the derivative of a constant is 0. So that's why we disregard what this this first pieces. And if we did the derivative it will undo the anti derivative. So the final answer will just be replacing X. In for those teas. And hopefully I've done some justice as to explaining instead of just saying All you need to do is replace the teas with X's natural final answer. But that's basically what this answer is. Um and hopefully I kind of explain why we have a different avenue to get that answer.

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