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If $ f $ is continuous on $ \mathbb{R} $, prove that $$ \int^b_a f(x + c) \, dx = \int^{b + c}_{a + c} f(x) \, dx $$ For the case where $ f(x) \ge 0 $, draw a diagram to interpret this equation geometrically as an equality of areas.
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01:31
Frank Lin
Calculus 1 / AB
Chapter 5
Integrals
Section 5
The Substitution Rule
Integration
Campbell University
Oregon State University
University of Nottingham
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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If $ f $ is continuous on …
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Prove that if a function $…
we know that for this question, withdrawing the diagram The first thing you could do a substitute you is expose. See, this indicates that D axe is do you? Because it's just one as the derivative of X. Therefore, in of the limits of integration, change to a pussy on the bottom and people see on the top after vax d axe Remember wise half of Expo See you had shifted by C units in the left direction horizontally after vax on the right and affects was seeing left.
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