by taking integral we get
integral sec(x) (tan(x)+sec(x)) dx
applying integral we get
sec(x) (tan(x)+sec(x)) gives sec^2(x)+tan(x) sec(x)
= integral (sec^2(x)+tan(x) sec(x)) dx Integrate the sum term by term
= integral sec^2(x) dx+ integral tan(x) sec(x) dx For the integrand tan(x) sec(x), now we will use substitution
substitute u = sec(x) and du = tan(x) sec(x) dx
= integral 1 du+ integral sec^2(x) dx The integral of sec^2(x) is tan(x)
= integral 1 du+tan(x) The integral of 1 is u
= u+tan(x)+constant
Substitute the value of u which is equal to
= sec(x):
so our conclusion is
:tan(x)+sec(x)+constant
hope this helps