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Integration of Sec 3x
The integration of sec 3x is given by (1/3) ln(tan 3x + sec 3x) + C, where C is the integration constant. It can be calculated using the substitution method. Mathematically, we can write the integration of sec 3x as ∫sec 3x dx = (1/3) ln(tan 3x + sec 3x) + C. Integration is the reverse process of differentiation, hence determining the integration of sec 3x is the same as finding the antiderivative of sec 3x.
Further in this article, we will evaluate the integration of sec 3x stepwise using the substitution method. We will also find the integration of sec 3x combined with different functions and solve a few examples for a better understanding.
1.  What is Integration of Sec 3x? 
2.  Integration of Sec 3x Formula 
3.  Integration of Sec 3x Using Substitution Method 
4.  FAQs on Integration of Sec 3x 
What is Integration of Sec 3x?
The integration of sec 3x is the process of finding the antiderivative of sec 3x. Mathematically, we can write the integration of sec 3x as ∫sec 3x dx = (1/3) ln(tan 3x + sec 3x) + C, where ∫ indicates the sign of integral, dx denotes that the integration of sec 3x is with respect to x, and C is the integration constant. We can calculate the integration of sec 3x using the substitution method by substituting 3x with another variable and solving the integral. Let now go through the formula for the integral of sec 3x.
Integration of Sec 3x Formula
The formula for the integration of sec 3x can be written as ∫sec 3x dx = (1/3) ln(tan 3x + sec 3x) + C with C as the constant of integration. The image given below shows the integral of sec 3x formula:
Integration of Sec 3x Using Substitution Method
Now that we know the integration of sec 3x as (1/3) ln(tan 3x + sec 3x) + C. We will prove this using the substitution method. For this assume 3x = u, differentiating both sides, we have 3 dx = du ⇒ dx = (1/3) du. We will use the derivative formulas of tan A and sec A which are:
 d(tan A)/dA = sec A tan A
 d(sec A)/dA = sec^{2}A
Simplifying the integral, we get
∫ sec 3x dx = ∫ sec u (1/3) du
= (1/3) ∫ sec u du
= (1/3) ∫ sec u (tan u + sec u) / (tan u + sec u) du [Multiplying and dividing by (tan u + sec u)]
= (1/3) ∫ (sec u tan u + sec^{2}u) / (tan u + sec u) du  (1)
Assume (tan u + sec u) = v ⇒ dv = (sec u tan u + sec^{2}u) du
Substituting in (1), we have
(1/3) ∫ (sec u tan u + sec^{2}u) / (tan u + sec u) du = = (1/3) ∫ (1 / v) dv
= (1/3) ln v + C
= (1/3) ln tan u + sec u + C
= (1/3) ln tan 3x + sec 3x + C
Hence, we have proved that the integration of sec 3x is (1/3) ln tan 3x + sec 3x + C.
Important Notes on Integration of Sec 3x:
 The formula for the integration of sec 3x can be written as ∫sec 3x dx = (1/3) ln(tan 3x + sec 3x) + C.
 We can calculate the integration of sec 3x using the substitution method by substituting 3x with another variable.
☛ Related Topics:
Integration of Sec 3x Examples

Example 1: Find the integration of sec 3x cos 3x.
Solution: To find the integration of sec 3x cos 3x, we will first simplify the expression.
sec 3x cos 3x = (1/cos 3x) cos 3x
= 1
∫sec 3x cos 3x dx = ∫1 dx
= x + C
Answer: Hence the integration of sec 3x cos 3x is x + C.

Example 2: Evaluate the integration of sec 3x tan 3x.
Solution: We know that the derivative of sec A is sec A tan A. Using this formula, we have a formula ∫ sec ax tan ax dx = (1/a) sec ax + C. Hence, we have
∫sec 3x tan 3x dx = (1/3) sec 3x + C
Answer: Therefore, the integration of sec 3x tan 3x is (1/3) sec 3x + C
FAQs on Integration of Sec 3x
What is Integration of Sec 3x?
The integration of sec 3x is the process of finding the antiderivative of sec 3x. Mathematically, it is given by (1/3) ln(tan 3x + sec 3x) + C, where C is the integration constant.
What is the Formula of Integration of Sec 3x?
The formula for the integration of sec 3x can be written as ∫sec 3x dx = (1/3) ln(tan 3x + sec 3x) + C with C as the constant of integration.
How to Find Integration of Sec 3x?
We can find the integration of sec 3x using the substitution method by substituting 3x with another variable and solving the integral.
What is the Integration of Sec^{3}x?
The integration of sec cube x is given by, ∫sec^{3}x dx = (1/2) sec x tan x + (1/2) ln sec x + tan x + C.
How to Determine the Integration of Sec 3x Tan 3x?
The integration of sec 3x tan 3x can be determined using the formula ∫ sec ax tan ax dx = (1/a) sec ax + C. We can substitute a = 3 in this formula and find the integration of sec 3x tan 3x as ∫sec 3x tan 3x dx = (1/3) sec 3x + C.
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