How do you Integrate $cot x d x$ $cotxdx$ by using substitution?

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How do you Integrate $cot x d x$ $cotxdx$ by using substitution?

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Answer 1 · 2015-02-26T15:06:08

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

$\int (\frac{1}{w}) d w = ln (| w |) + C$ $int (1/w) dw = ln(|w|) + C$

$\frac{d sin (x)}{d x} = cos (x)$ $( d sin(x))/(dx) = cos(x)$

O.K. here we go:
$\int (cot (x)) d x = \int (\frac{cos (x)}{sin (x)}) d x$ $int (cot(x)) dx = int ( (cos(x))/(sin(x)) ) dx$

If we let
$w = sin (x)$ $w = sin(x)$ then $\frac{d w}{d x} = cos (x)$ $(dw)/(dx) = cos(x)$
so
$\int \frac{cos (x)}{sin (x)} d x = \int (\frac{\frac{d w}{d x}}{w}) d x$ $int (cos(x))/(sin (x)) dx = int ( ((dw)/(dx))/w) dx$

= $\int \frac{1}{w} d w$ $int 1/w dw$

$= ln (| w |) + C$ $= ln(|w|) + C$

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

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How do you Integrate $cot x d x$ $cotxdx$ by using substitution?

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How do you Integrate $cot x d x$ $cotxdx$ by using substitution?