integrate: dx/tanx+cotx+secx+cosecx
Asked by Akshansh prasad | 17th Sep, 2013, 03:29: PM
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
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 | 19th Sep, 2013, 10:03: AM
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