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integrate: dx/tanx+cotx+secx+cosecx

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

Answered by Expert 19th September 2013, 10:03 AM
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