# Homework Help: Indefinite integrals with different solutions?

1. Mar 18, 2014

### AntSC

Indefinite integrals with different solutions???

1. The problem statement, all variables and given/known data

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

2. Relevant equations

As above

3. 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. Mar 18, 2014

### Dick

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. Mar 18, 2014

### arildno

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.

4. Mar 18, 2014

### AntSC

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. Mar 18, 2014

### arildno

As long as your purported antiderivatives are TRUE antiderivatives, then they do only differ by at most a non-zero constant.

6. Mar 18, 2014

### AntSC

Got it. Thanks very much :)