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Integrate the functions1/(1-tanx)...

Integrate the functions`1/(1-tanx)`

Text Solution

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let `I = int 1/(1-tanx) dx`
`I = int 1/(1-sinx/cosx) dx`
=` int 1/((cosx-sinx)/cosx) dx`
= `int cosx/(cosx - sinx ) dx`
=`1/2 int (2cosx)/(cosx - sinx ) dx`
= `1/2 int( cosx+ cosx)/(cosx - sinx ) dx`
Adding and subtracting `sinx` in the numerator
= `1/2 int(( cosx- sinx)(sinx+cosx))/(cosx - sinx ) dx`
= `1/2 int(( cosx- sinx)/(cosx - sinx ) + (sinx+cosx)/(cosx - sinx )) dx`
= `1/2 { int 1 dx - int (-sinx- cosx)/(cosx - sinx ))} dx`
= `1/2{ x - log|cosx- sinx|} + c`
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