Integrating Cosecant

Calculus Level 3

$\begin{aligned} I_1 &=&\int \csc x \, dx = \int \dfrac{\csc x (\csc x - \cot x)}{\csc x - \cot x} \, dx = \ln | \csc x - \cot x | \\ I_2 &=&\int \csc x \, dx = \int \dfrac{-\csc x (\csc x + \cot x)}{\csc x + \cot x} \, dx = - \ln | \csc x + \cot x | \\ \end{aligned}$

If both calculations above are correct, then $I_1$ and $I_2$ differ by a constant. Find $I_1 -I_2$ .

The answer is 0.

2 solutions

Aditya Narayan Sharma
May 20, 2016

Simply we see $I_1=I_2$ , It's obvious $\displaystyle \int cscx dx - \int cscx dx=0$

The difference of two integrals of the same function is a constant, not necessarily a zero. Your "solution" is not correct.

Marta Reece - 5 years ago

As you can see from the problem the constant "C" is not mentioned too. According to the problem we may have the two integrals equal.

Aditya Narayan Sharma - 5 years ago

See my addition to solution for an example of why it's not obvious that two integrals of the same function are the same.

Marta Reece - 5 years ago

@Marta Reece – That's a nice bit of work I admire, The fact is $\displaystyle \int cosecx dx = ln|cosecx-cotx|+C$ , but the problem statement has assumed C to be zero. Check your problem there's no mention of a constant when you integrate the function in two different forms. It is zero only since your problem makes it seem zero else we can't define such a difference between two integrals as their answer is a family of solution.

Aditya Narayan Sharma - 5 years ago

@Aditya Narayan Sharma – With the constant, and the constant being arbitrary, every correct solution is equal to any other correct solution. Therefore I had to leave the constant out. That means I was posting a particular, not a general, solution. Particular solutions may differ by a constant. I was asking what this constant is in this particular case. It happens to be a zero, but that was not a foregone conclusion. It did not follow from the fact that the constant of integration was left out.

Marta Reece - 5 years ago

Marta Reece
May 19, 2016

$\begin{aligned} I_1 - I_2 &=& \ln |\csc x - \cot x| + \ln |\csc x + \cot x | = \ln |(\csc x - \cot x)(\csc x + \cot x) | \\ &=& \ln |(\csc^{2} x - \cot^{2} x) |= \ln 1 = 0 \\ \end{aligned}$

The solution proper ends here. What follows is for those who do not believe that two integrals of the same function may be different. The text has been placed here only because I am not able to put a picture into a comment.

Here are they graphed.

Integrating Cosecant

The answer is 0.

2 solutions

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