Indefinite Integration of sec x
How to find the indefinite integration of sec x: example and its solution.
Multiply cos x
to both of the numerator and the denominator.
Change the function to make (constant)/(linear) form
by using partial fraction decomposition.
Set 1/(1 + t)(1 - t) = A/(1 - t) + B/(1 - t)
and compare both sides' numerators
to find A and B.
A = B = 1/2
So 1/(1 + t)(1 - t) = (1/2)(1/[1 - t]) + (1/2)(1/[1 - t])
= (1/2)(1/[1 - t] + 1/[1 - t]).
So (given) = ∫ (1/2)(1/[1 - t] + 1/[1 - t]) dt.
Put sin x in t-s.
Then (given) = (1/2) ln |(1 + sin x)/(1 - sin x)| + C.
This is the answer.
But let's go further to see the practical formula.
Multiply (1 + sin x)/(1 + sin x).
(1 - sin x)(1 + sin x) = 1 - sin2 x
Product of a sum and a difference
Take the square exponents
out from both of the numerator and the denominator.
Power of a quotient
Take the exponent out from the ln.
Logarithms of powers
Split the fraction into two parts.
1/cos x = sec x
sin x/cos x = tan x
So ∫ sec x dx = ln |sec x + tan x| + C.
This answer is the practical formula
that is widely used in various fields.