What is the integral of sec3(x)?
Integration is the process of finding the limit of the total number of elements when the number of elements tends to be infinite and at the same time, each term becomes small.
Answer. The integral of sec3(x) is 1/2 [(sec x tan x + ln |sec x + tan x| ] + C
Let us see, how we can find the required integral.
We will use the product rule to find the integral.
We can write sec3x as
⇒ sec3 x = sec (x) × sec2 (x)
∫sec3 x dx = ∫ sec (x) × sec2 (x) dx
Let u = sec (x) and v = sec2 (x) dx
∫uv = u ∫v dx - ∫ (du / dx ∫v dx) dx. [ from product rule of integration ] ------(1)
⇒ ∫v dx = tan x [ from integration formulae ]
⇒ du / dx = sec (x) tan (x)
Substituting these values in equation 1 we get,
⇒ ∫sec3 x dx = (sec x) (tan x) - ∫ (sec x tan x) tan x dx ------(2)
From Trigonometric identities,
tan2 x = sec2 x – 1
On substituting the above value in equation (2), we get
⇒ ∫sec3 x dx = sec (x) tan (x) – ∫ (sec 2x – 1) sec (x) dx. [ from identity]
⇒ ∫sec3 x dx = sec (x) tan (x) – ∫sec 3(x) + sec (x) dx [ opening bracket by multiplying with minus sign]
⇒ 2 ∫sec3 x dx = sec (x) tan (x) + ∫ sec (x) dx. [ rearranging terms by shifting ∫sec 3(x) on LHS]
⇒ ∫ sec3 (x) =1/2 [ (sec (x) tan (x) + ln sec (x) + tan (x) | ]+ C. [ from integration formulae ∫ sec (x) dx = ln│sec (x) + tan (x) | ]
Therefore, ∫ sec3 (x) =1/2 [ (sec (x) tan (x) + ln│sec (x) + tan (x) | ]+ C