What is the integral of sec3(x)?
Integration is the process of adding or summing up the parts to find the whole. It finds the area under the curve.
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. [ Using Integration by Parts ] ------(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
⇒ ∫sec3 x dx = sec (x) tan (x) – ∫sec 3(x)dx + ∫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