How do you integrate #cscx #?

Answer 1

# int \ csc x \ dx = - ln|csc(x) + cot(x)| +C #

There are many ways to prove this result. The quickest method that I am aware of is as follows:

# int \ csc x \ dx = int \ cscx \ (cscx + cotx)/(cscx + cotx) \ dx # # " "= int \ (csc^2x + cscxcotx)/(cscx + cotx) \ dx #

Then we perform simple substitution, Let

#u = cscx + cotx => (du)/dx = -cscxcotx - csc^2x # # " "= -(cscxcotx + csc^2x) #

And so:

# int \ csc x \ dx = int \ (-1/u) \ du # # " "= - int \ 1/u \ du # # " "= - ln|u| +C # # " "= - ln|cscx + cotx| +C #

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Answer 2

Ezekiel Alexander

To integrate csc(x), you use the integral of its reciprocal, which is -ln|csc(x) + cot(x)| + C.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.