How do you integrate $cscx$ ?

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Answer 1 · 2017-02-18T22:49:31

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

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:

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

How do you integrate cscx cscx ?