How Do You Integrate (16x^4 - 4)/(4x^2+1) from 0 to 1?

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Homework Help Overview

The problem involves evaluating the definite integral of the function (16x^4 - 4)/(4x^2 + 1) from 0 to 1. The subject area pertains to calculus, specifically integration techniques.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss potential methods such as polynomial division and substitution. There is a question regarding the correctness of the denominator, with one participant suggesting it might be 4x^2 + 2 instead of 4x^2 + 1. Others mention the factorization of the numerator and the implications for integration.

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants have offered hints and suggestions, such as using polynomial division and considering substitution, but there is no explicit consensus on the best method to proceed.

Contextual Notes

Participants are navigating potential misunderstandings regarding the integrand's structure, particularly the denominator, which may affect the integration process. There is also a mention of the complexity of the problem, indicating that it may not yield straightforward solutions.

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Homework Statement



integral defined in 0 (down) and 1 (up) of:
(16x^4 - 4)/(4x^2+1)

Homework Equations





The Attempt at a Solution



maybe the partition mode?
who help me?
 
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Just wondering...by any chance, is the denominator of the integrand 4x2+2?
 
Last edited:
Hint:
Use polynomial division first.
 
i can use the substitution rule, with u=4x^2?
 
You could but it doesn't really help since du= 8xdx doesn't give you anything easy. The numerator obviously factors into (4x2-2)(4x2+2)- that's why arildno asked if the denominator wasn't actually 4x2+ 2 rather than 4x2+ 1. But the world is never that easy, not even homework problems.

Best thing to do is arildno's suggest. Go ahead and divide 16x4- 4 by 4x2+ 1. The result will be a cubic polynomial plus a linear term, Ax+ B, over 4x2+ 1. To integrate Ax/(4x2+1), let u= 4x2+ 1. To integrate B/(4x2+1), use the arctangent.
 

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