This is done using Integration by Parts
int u*dv=uv-int v*du∫u⋅dv=uv−∫v⋅du
Let u=sec xu=secx
Let dv=sec^2 x*dxdv=sec2x⋅dx
Let v=tan xv=tanx
Let du=sec x*tan x* dxdu=secx⋅tanx⋅dx
Use the formula
int u*dv=uv-int v*du∫u⋅dv=uv−∫v⋅du
int sec x*sec^2 x*dx=sec x*tan x-int tan x(sec x*tan x* dx)∫secx⋅sec2x⋅dx=secx⋅tanx−∫tanx(secx⋅tanx⋅dx)
int sec^3 x*dx=sec x*tan x-int tan^2 x*sec x* dx∫sec3x⋅dx=secx⋅tanx−∫tan2x⋅secx⋅dx
Recall tan^2 x+1=sec^2 xtan2x+1=sec2x
and tan^2 x=sec^2 x-1tan2x=sec2x−1
int sec^3 x*dx=sec x*tan x-int (sec^2 x-1)sec x* dx∫sec3x⋅dx=secx⋅tanx−∫(sec2x−1)secx⋅dx
int sec^3 x*dx=sec x*tan x-int (sec^3 x-sec x)* dx∫sec3x⋅dx=secx⋅tanx−∫(sec3x−secx)⋅dx
int sec^3 x*dx=sec x*tan x-int sec^3 x*dx+int sec x* dx∫sec3x⋅dx=secx⋅tanx−∫sec3x⋅dx+∫secx⋅dx
Transpose the right int sec^3 x*dx∫sec3x⋅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∫sec3x⋅dx+∫sec3x⋅dx=secx⋅tanx+∫secdx
2*int sec^3 x*dx=sec x*tan x+ln(sec x+tan x)2⋅∫sec3x⋅dx=secx⋅tanx+ln(secx+tanx)
Divide both sides by 22
color(red)(int sec^3 x*dx=1/2*sec x*tan x+1/2*ln(sec x+tan x)+C)∫sec3x⋅dx=12⋅secx⋅tanx+12⋅ln(secx+tanx)+C
God bless....I hope the explanation is useful.
