Indefinite integrals with different solutions?

In summary, when solving indefinite integrals using pattern recognition, it is important to remember that two functions that differ by a constant can both be valid forms for the indefinite integral. This is due to the identity between cosine and sine. So if different solutions are obtained, the difference is likely due to constants.
  • #1
AntSC
65
3
Indefinite integrals with different solutions?

Homework Statement



[tex]\int \csc ^{2}2x\cot 2x\: dx[/tex]
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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  • #2
AntSC said:

Homework Statement



[tex]\int \csc ^{2}2x\cot 2x\: dx[/tex]
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.
 
  • #3
Remember that:
[tex]\frac{\cos^{2}(y)}{\sin^{2}(y)}=\frac{1}{\sin^{2}(y)}-1[/tex]
due to the age-oldest relation between cos and sin. :smile:
 
  • #4
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?
 
  • #5
As long as your purported antiderivatives are TRUE antiderivatives, then they do only differ by at most a non-zero constant. :smile:
 
  • #6
Got it. Thanks very much :)
 

FAQ: Indefinite integrals with different solutions?

1. What is an indefinite integral?

An indefinite integral is the reverse process of differentiation. It involves finding a function whose derivative is equal to the given function.

2. What are the different methods for solving indefinite integrals?

Some common methods for solving indefinite integrals include substitution, integration by parts, trigonometric substitution, and partial fractions.

3. Can an indefinite integral have multiple solutions?

Yes, an indefinite integral can have multiple solutions. This is because when finding the antiderivative, a constant of integration is added, resulting in an infinite number of possible solutions.

4. How do I know which solution is correct for an indefinite integral?

To determine the correct solution for an indefinite integral, you can check your answer by taking the derivative and verifying that it is equal to the original function. You can also use initial conditions or given information to narrow down the possible solutions.

5. Are there any tips for solving indefinite integrals?

Some tips for solving indefinite integrals include recognizing patterns, using substitution to simplify the integrand, and practicing regularly to improve your skills. It is also helpful to have a good understanding of basic integration rules and techniques.

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