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# What is the integral of sec^{3}(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 sec^{3}x as

⇒ sec^{3 }x = sec (x) × sec^{2 }(x)

∫sec^{3 }x dx = ∫ sec (x) × sec^{2 }(x) dx

Let u = sec (x) and v = sec^{2 }(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,

⇒ ∫sec^{3 }x dx = (sec x) (tan x) - ∫ (sec x tan x) tan x dx ------(2)

From Trigonometric identities,

tan^{2 }x = sec^{2} x – 1

On substituting the above value in equation (2), we get

⇒ ∫sec^{3 }x dx = sec (x) tan (x) – ∫ (sec ^{2}x – 1) sec (x) dx

⇒ ∫sec^{3 }x dx = sec (x) tan (x) – ∫sec ^{3}(x)dx + ∫sec (x)dx [ opening bracket by multiplying with minus sign]

⇒ 2 ∫sec^{3 }x dx = sec (x) tan (x) + ∫ sec (x) dx. [ rearranging terms by shifting ∫sec ^{3}(x) on LHS]

⇒ ∫ sec^{3} (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, ∫ sec^{3} (x) =1/2 [ (sec (x) tan (x) + ln│sec (x) + tan (x) | ] + C

## What is the integral of sec^{3}(x)?

**Summary:**

The integral of sec^{3}(x) is 1/2 [(sec x tan x + ln |sec x + tan x| ] + C

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