[Math] Why do different substitutions give different results for $\int\cot(x)\csc(x)^2\,dx?$


I am having some issues when trying to integrate this function. First of all I have to decide to make a $U$ substitution either for $\csc(x)$ or for $\cot(x)$, both of them are acceptable substitutions, I believe. Now, when I solve the integral after doing a $U$ substitution for $\csc(x)$ I get a different answer than when I do a $U$ substitution for $\cot(x)$. After observing that the answers were not matching I decided to rely on technology. I used my TI-89 calculator that gave me a different answer than when I do it on wolfram-alpha.

Meaning that I am back to the beginning. My question is which is the right answer? Why are these two reliable calculators giving different answers?

$\frac{-\csc(x)^2}{2}$ and $\frac{-\cot(x)^2}{2}$
are the two answers I got.

Best Answer

The reason why you are getting different answers is the importance of the $+C$. Read this: https://brilliant.org/discussions/thread/the-importance-of-c/

To answer your answer, both are right.

Also note that:

$$\cot^2(x)=\csc^2(x)-1$$ and as $1$ is just constant, it is accounted for in the $+C$, merely the $C$ in the two forms are different.

Edit: $$\frac{-\cot^2(x)}{2}=\frac{-\csc^2(x)+1}{2}=\frac{-\csc^2(x)}{2}+\frac{1}{2}$$ Now what is $\frac{1}{2}+C$. It can be made into another $C$. And you are left with $$-\frac{\csc^2(x)}{2}+C$$

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