Solve Value of Integration: Find Integral from 0 to 2

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Homework Help Overview

The problem involves finding the integral of a function defined in terms of itself, specifically f(x) = x^2 - x - ∫₀¹ f(x) dx, and requires evaluating ∫₀² f(x) dx.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss integrating both sides of the equation defining f to find the integral from 0 to 1. There are attempts to express the integral from 0 to 2 in terms of known quantities and constants, with some questioning the possibility of obtaining a numerical value for the integral.

Discussion Status

Participants are exploring different methods to express the integral and are providing insights into how to approach the problem. There is an acknowledgment of the potential to find numerical values, but no consensus has been reached on a specific method or solution.

Contextual Notes

There is an implicit assumption that the integral from 0 to 1 can be evaluated, and participants are navigating the implications of defining f in terms of its own integral.

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Homework Statement


If f(x) = x^2 - x - \int_0^1 f(x) dx, find \int_0^2f(x) dx


Homework Equations





The Attempt at a Solution


I found \int_0^2f(x) dx = \frac{2}{3} - 2 \int_0^1 f(x) dx

Is it possible the answer in numerical value? If yes, please guide me. Thanks
 
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Yes, it is. In fact, you can find the integral of f from 0 to 1 just by integrating both sides of the equation defining f.
 
The definite integral is a number: \int_0^1{f(x)dx}=A
The first equation can be written as f(x)=x^2-x-A.
Integrate it from x=0 to x=1: you get an equation for A.

ehild
 
I managed to get something slightly prettier by writing
\int_0^2 f(x) dx = I + \int_1^2 (x^2 - x - I) dx
where
I = \int_0^1 f(y) dy
is a constant.
 
Ah I get it. Integrating f(x) from 0 to 1 never crosses my mind. Thanks a lot
 

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