What we have here as given is
#int (x* sec x^2) dx#
take note: differential of #x^2# is #d(x^2)=2x*dx#
We do a numerical preparation here by placing a 2 inside the integral and 1/2 outside the integral.
#int (x* sec x^2) dx=1/2int ( sec x^2)*(2x) dx#
from the formula
#int sec u* du=ln (sec u+tan u)+C#
#color(red)(int (x* sec x^2) dx=1/2int ( sec x^2)*(2x) dx=#
#color(red)(1/2*ln (sec x^2+tan x^2)+C)#
Let us do a little checking by differentiation
from our answer
#1/2*ln (sec x^2+tan x^2)+C#
Let us perform differentiation
#d/dx(1/2*ln (sec x^2+tan x^2)+C)#
#d/dx(1/2*ln (sec x^2+tan x^2))+d/dx(C)#
#1/2*(1/(sec x^2+tan x^2))*d/dx(sec x^2+tan x^2)+0#
#1/2*(1/(sec x^2+tan x^2))*(sec x^2*tan x^2*d/dx(x^2)+sec^2 x^2*d/dx(x^2))+0#
#1/2*(1/(sec x^2+tan x^2))*(sec x^2*tan x^2*2x+sec^2 x^2*2x)#
factoring the common monomial factor #2x*sec x^2#
#1/2*((2x*sec x^2)/(sec x^2+tan x^2))*(sec x^2+tan x^2)#
#1/cancel2*((cancel2x*sec x^2)/cancel(sec x^2+tan x^2))*cancel(sec x^2+tan x^2)#
#x * sec x^2" "#which is the given integrand
God bless....I hope the explanation is useful.
