How to Solve Tricky Integrals Involving x^3*sin(x^2) and Rational Functions?

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Homework Help Overview

The discussion revolves around solving integrals involving the function x^3 * sin(x^2) and a rational function represented as (4x + 2) / (x^2(x^2 + 2)). Participants are exploring different integration techniques to tackle these problems.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to use integration by parts for the first integral but expresses difficulty in proceeding. Some participants suggest a substitution to simplify the integration process, while others discuss breaking the product into components for integration by parts. For the second integral, a participant introduces the concept of Partial Fraction Decomposition and suggests that each resulting fraction could be integrated using substitution.

Discussion Status

The discussion is active, with participants offering various approaches and substitutions for the integrals. There is no explicit consensus on a single method, but several productive lines of reasoning are being explored.

Contextual Notes

Participants are working within the constraints of homework rules, which may limit the extent of guidance provided. There is an emphasis on understanding the setup and assumptions behind the integrals being discussed.

Lodve
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Hi I'm struggling solving this integration:
x^3 * sin x^2
My idea to solve this one is to apply integration by parts, but I can't get through using it.

\frac{4x+2}{x^2(x^2+2)} this seems a hard nut to crack.
 
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If you want to use integration by parts for the first one, there is a certain substitution that would make it quite simple. Can you see it?
 


Hmmm not sure, what if a let the u equal to x^2 and derive it so that I get 2x?
 


For integration by parts, I would start by breaking the product up into x2 and xsin(x2).
 


u=x^2
du=dx*2x

so your argument would be:

1/2u*sin(u)du

which is easier to do parts on.
 


For the second integral, use Partial Fraction Decomposition:

<br /> \frac{4x+2}{x^2(x^2+2)} = \frac{Ax+B}{x^2}+\frac{Cx+D}{(x^2+2)}<br />

Each of the resulting fraction could be integrated by u, du substitution.

http://en.wikipedia.org/wiki/Partial_fraction
 

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