Solving this definite integral using integration by parts

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Homework Help Overview

The discussion revolves around solving a definite integral using integration by parts, specifically focusing on the integral denoted as \(I_n\). The subject area is integral calculus, with an emphasis on techniques such as integration by parts and manipulation of integrals.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants explore the relationship between \(I_n\) and \(I_{n+1}\), considering how to express the integral in a different form. There are hints provided regarding manipulating the expression for \(x^2\) to facilitate the integration process.

Discussion Status

The discussion is ongoing, with participants sharing various approaches and hints. Some guidance has been offered, particularly in terms of manipulating the integral and considering relationships between different forms of the integral.

Contextual Notes

Participants are working within the constraints of a homework problem, which may limit the information available or the methods that can be employed. There is an emphasis on exploring different representations of the integral without arriving at a final solution.

songoku
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Homework Statement
Please see below
Relevant Equations
Integration
1684334479139.png


Using integration by parts:
$$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$
$$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$
Then how to continue?

Thanks
 
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My first approach would be work out what In+1 is and see how they relate.

Their hint of multiplying by 1 tells me that I might need to represent 1 as a fraction where numerator = denominator.
 
Last edited:
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Hint: ##x^2 = (x^2+1) - 1##.
 
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I understand

Thank you very much scottdave and vela
 
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