Trig Identity Integral Homework: Solving a Tricky Equation Using Substitutions

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Homework Help Overview

The discussion revolves around evaluating the integral \(\int^3_0\frac{1+\arctan(\frac{x}{3})}{9+x^2}dx\), with a focus on trigonometric identities and integration techniques. Participants are exploring the necessary approaches to tackle this integral, particularly in the context of trigonometric functions and their derivatives.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about the integral and suggests that it may involve a trigonometric identity. Some participants propose splitting the integral into two parts, indicating that one part can be solved directly while the other may require substitution. There is also a mention of the derivative of the arctangent function and its relevance to the integral.

Discussion Status

The discussion is ongoing, with participants offering various insights and suggestions regarding the approach to the integral. There is an exploration of different methods, including substitution and differentiation, but no consensus has been reached on a definitive solution or method.

Contextual Notes

Participants note the original poster's missed class on trigonometric identities, which may contribute to their uncertainty. The discussion includes references to differentiation formulas and the implications of certain mathematical identities, indicating a need for clarification on these concepts.

theRukus
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Homework Statement


I missed one class on trigonometric identities in integrals, and I feel that one is needed here:

\int^3_0\frac{1+arctan(\frac{x}{3})}{9+x^2}dx

Homework Equations


The Attempt at a Solution


Again, I'm unsure what to do. I think that it is a trig identity, but I could be wrong. I'll continue to try parts & substitutions..

Thanks!
 
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The derivative of acrtan(x) is 1/(x2+1), so the derivative of acrtan(x/3) is 3/(x2+9)
 
theRukus said:

Homework Statement


I missed one class on trigonometric identities in integrals, and I feel that one is needed here:

\int^3_0\frac{1+arctan(\frac{x}{3})}{9+x^2}dx


Homework Equations





The Attempt at a Solution


Again, I'm unsure what to do. I think that it is a trig identity, but I could be wrong. I'll continue to try parts & substitutions..

Thanks!
Split the integral into two integrals. The first can be done directly and the second can be done with an ordinary substitution. Integration by parts is not the way to go.

Do you know this differentiation formula?
\frac{d}{dx} tan^{-1}(x)?
 
Notice that \displaystyle \frac{d((f(x)^2)}{dx}=2f(x)f'(x)\,.

That does that imply regarding \displaystyle \int\ f(x)f\,'(x)\,dx\,?
 

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