This is done using Integration by Parts
#int u*dv=uv-int v*du#
Let #u=sec x#
Let #dv=sec^2 x*dx#
Let #v=tan x#
Let #du=sec x*tan x* dx#
Use the formula
#int u*dv=uv-int v*du#
#int sec x*sec^2 x*dx=sec x*tan x-int tan x(sec x*tan x* dx)#
#int sec^3 x*dx=sec x*tan x-int tan^2 x*sec x* dx#
Recall #tan^2 x+1=sec^2 x#
and #tan^2 x=sec^2 x-1#
#int sec^3 x*dx=sec x*tan x-int (sec^2 x-1)sec x* dx#
#int sec^3 x*dx=sec x*tan x-int (sec^3 x-sec x)* dx#
#int sec^3 x*dx=sec x*tan x-int sec^3 x*dx+int sec x* dx#
Transpose the right #int sec^3 x*dx# to the left side of the equation
#int sec^3 x*dx+int sec^3 x*dx=sec x*tan x+int sec dx#
#2*int sec^3 x*dx=sec x*tan x+ln(sec x+tan x)#
Divide both sides by #2#
#color(red)(int sec^3 x*dx=1/2*sec x*tan x+1/2*ln(sec x+tan x)+C)#
God bless....I hope the explanation is useful.
