Indefinite integrals with different solutions?

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Homework Help Overview

The discussion revolves around the evaluation of the indefinite integral \(\int \csc^{2}2x\cot 2x\: dx\) with a focus on recognizing patterns rather than using substitution. Participants are exploring why two different functions, which both differentiate to the same integrand, yield different forms of the integral's solution.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss attempting to find functions that, when differentiated, match the integrand. There is a question regarding the significance of obtaining different solutions that appear similar. Some participants reflect on the nature of constants in indefinite integrals and how they relate to the solutions.

Discussion Status

The conversation has led to some productive insights regarding the relationship between different antiderivatives and the role of constants. Participants are engaging with the concept that differing solutions may simply reflect differences in constant terms, although there is no explicit consensus on all aspects of the discussion.

Contextual Notes

Participants note the importance of recognizing identities in trigonometric functions, which may influence their understanding of the integral's solutions. There is an acknowledgment of the potential for confusion when differentiating between similar functions.

AntSC
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Indefinite integrals with different solutions?

Homework Statement



\int \csc ^{2}2x\cot 2x\: dx
Solve without substitution using pattern recognition

Homework Equations



As above

The Attempt at a Solution



To try a function that, when differentiated, is of the same form as the integrand.
The two solutions are attached.
My question is that both functions i tried, differentiated to the integrand, but as a result they yield a different solution to the integral. These two functions look very similar. I don't understand what the significance of this is. I've seen a similar result when playing around with other integrals. Any light on this would be really helpful. Cheers.
 

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AntSC said:

Homework Statement



\int \csc ^{2}2x\cot 2x\: dx
Solve without substitution using pattern recognition

Homework Equations



As above

The Attempt at a Solution



To try a function that, when differentiated, is of the same form as the integrand.
The two solutions are attached.
My question is that both functions i tried, differentiated to the integrand, but as a result they yield a different solution to the integral. These two functions look very similar. I don't understand what the significance of this is. I've seen a similar result when playing around with other integrals. Any light on this would be really helpful. Cheers.

cosec^2=cot^2+1. Two functions that differ by a constant have the same derivative - so both are fine forms for the indefinite integral.
 
Remember that:
\frac{\cos^{2}(y)}{\sin^{2}(y)}=\frac{1}{\sin^{2}(y)}-1
due to the age-oldest relation between cos and sin. :smile:
 
Aha! Of course! Totally forgot that. I was looking at differences in constants but completely missed the identity.
So in general, if solutions to an integral are different (if one is on the ball to spot it) then the difference is due to the constants?
 
As long as your purported antiderivatives are TRUE antiderivatives, then they do only differ by at most a non-zero constant. :smile:
 
Got it. Thanks very much :)
 

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