How do you Integrate $cotxdx$ by using substitution?

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Answer 1 · 2015-02-26T15:06:08

Some basic identities that we'll need:
$cos(x) = (cos(x))/(sin(x))$

$int (1/w) dw = ln(|w|) + C$

$( d sin(x))/(dx) = cos(x)$

O.K. here we go:
$int (cot(x)) dx = int ( (cos(x))/(sin(x)) ) dx$

If we let
$w = sin(x)$ then $(dw)/(dx) = cos(x)$
so
$int (cos(x))/(sin (x)) dx = int ( ((dw)/(dx))/w) dx$

= $int 1/w dw$

$= ln(|w|) + C$

$= ln(|sin(x)|) + C$

How do you Integrate cotxdxcotxdx by using substitution?