# Thread: Integration problem with two solutions?

1. ## Integration problem with two solutions?

Here is the question: $\int \(csc ^{2}2x\cot 2x dx$ Find this integral by inspection.
Getting the answer they expect is simple enough, consider the derivative of $(\cot 2x)^{2}$ which is $-4\csc ^{2}2x\cot 2x$ and then adjust by a factor of $-\frac{1}{4}$

This gives an answer of $-\frac{1}{4}\cot ^{2}2x + C$

However if you adjust the first integral they give you by factoring out $\csc 2x$ and then apply the same technique with $(\csc 2x)^{2}$ instead the answer turns out to be - $\frac{1}{4}\csc ^{2}2x + C$. Both differentiate to be the same thing, (unless I'm making an obvious mistake so are they both correct?

Please feel free to correct my thinking or maths if it's flawed

2. What's going on here is that from the identity

$\sin^{2}(2x)+\cos^{2}(2x)=1,$ you may divide through by $\sin^{2}(2x)$ to obtain the identity

$1+\cot^{2}(2x)=\csc^{2}(2x).$

The different between $\cot^{2}(2x)$ and $\csc^{2}(2x)$ is $1,$ which can be absorbed by the constant of integration. Hence, either answer is correct. In a definite integral, either would work.

3. Thank You!!! I had considered that, but didn't know whether that could happen, thanks for the quick reply

4. You're very welcome. Have a good one!