Question
Tue September 17, 2013 By: Akshansh Prasad
 

integrate: dx/tanx+cotx+secx+cosecx

Expert Reply
Thu September 19, 2013

let A = Tanx+Cotx+Secx+Cosecx

     A = (Sinx/Cosx)+(Cosx/Sinx)+(1/Cosx)+(1/Sinx)

        = (Sinx^2+Cosx^2+Sinx+Cosx)/Sinx.Cosx

        = (1 + Sinx+Cosx)/Sinx.Cosx

Multiply by (Sinx+Cosx-1) in denominator and numerator

     A  = (Sinx+Cosx+1)(Sinx+Cosx-1) / (Sinx.Cosx)(Sinx+Cosx-1)

          = (Sinx^2+Cosx^2+2Sinx.Cosx-1) / (Sinx.Cosx)(Sinx+Cosx-1)

          = (1+2Sinx.Cosx-1) / (Sinx.Cosx)(Sinx+Cosx-1)     

          = (2Sinx.Cosx) / (Sinx.Cosx)(Sinx+Cosx-1)    

          = 2 / (Sinx+Cosx-1)  

    I     = integrate dx/(Tanx+Cotx+Secx+Cosecx)

    I    = integrate dx/A

    I    = integrate (Sinx+Cosx-1) dx/2

    I   =  (-Cosx + Sinx -x)/2 + C

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