Question

Evaluate : `int  "e"^(3"x")/("e"^(3"x") + 1)` dx

Answer 1

Let I = `"e"^(3"x")/("e"^(3"x") + 1)` dx

Put e^3x + 1 = t

Diff. both the sides w.r.t. x 

3 e^3x = dt ⇒  e^3x  dx = `"dt"/3`

`therefore "I" = 1/3 int  "dt"/"t"`

`= 1/3 "log" |"t"| + "c"`

`= 1/3 "log" |e^(3x) + 1| + "C"`