What is the integral of sec³(x)?

Solution :

Integration is the process of adding or summing up the parts to find the whole. It finds the area under the curve.

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

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What is the integral of sec³(x)?

Summary:

The integral of sec³(x) is 1/2 [(sec x tan x + ln |sec x + tan x| ] + C