Could someone please explain to me how to get from this firs

Could someone please explain to me how to get from this first equation to the second equation? I don\'t really need to know how to apply the limits of integration. I just need to know how to integrate this equation.

I need steps, please. Or an explanation as to how they got to the second equation. Maybe it\'s a known integral rule I can\'t remember. I\'m not sure.

Solution

Integral of (e^-x) sqrt(1+e^-2x) dx

=> Say (e^-x) = y

dy = - (e^-x) dx

=> Integral of - sqrt(1+y^2) dy

let y = tan u

dy = sec^2 u du

sqrt(1+ tan^2 u) =sec u and u=arctan y

=> Integral of -(sec^3 u du)

Use the reduction formula to get

Integral of [- sec^3 u du]

=> -[1/2 tan u sec u + 1/2 (Integral of sec u du)]

=> -[1/2 tan u sec u + 1/2 In(tan u + sec u)] + C

Substitute back for u = arctan y,

=> -[(1/2) y sqrt(y^2 + 1) + (1/2) In(sqrt(y^2 +1) + y)] + C

Substitute back for (e^-x) = y,

=> -[(1/2) (e^-x)sqrt(1+e^-2x) + (1/2) In(e^-x + sqrt(1+e^-2x))] + C