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Integral of Sec x
To find the integral of sec x, we will have to use some facts from trigonometry. Sec x is the reciprocal of cos x and tan x can be written as (sin x)/(cos x). We can do the integration of secant x in multiple methods such as:
 By using substitution method
 By using partial fractions
 By using trigonometric formulas
 By using hyperbolic functions
We have multiple formulas for integration of sec x and let us derive each of them using the above mentioned methods. Also, we will solve some examples related to the integral of sec x.
What is the Integral of Sec x?
The integral of sec x is lnsec x + tan x + C. It denoted by ∫ sec x dx. This is also known as the antiderivative of sec x. We have multiple formulas for this. But the more popular formula is, ∫ sec x dx = ln sec x + tan x + C. Here "ln" stands for natural logarithm and 'C' is the integration constant. Multiple formulas for the integral of sec x are listed below:
 ∫ sec x dx = ln sec x + tan x + C [OR]
 ∫ sec x dx = (1/2) ln  (1 + sin x) / (1  sin x)  + C [OR]
 ∫ sec x dx = ln  tan [ (x/2) + (π/4) ]  + C
 ∫ sec x dx = cosh^{1}(sec x) + C (or) sinh^{1}(tan x) + C (or) tanh^{1}(sin x) + C
We use one of these formulas according to necessity.
We will prove each of these formulas in different methods. Sounds interesting? Let's go!
Integral of Sec x by Substitution Method
We can find the integral of sec x by substitution method. For this, we multiply and divide the integrand with (sec x + tan x). But why do we need to do this? Let us see.
∫ sec x dx = ∫ sec x · (sec x + tan x) / (sec x + tan x) dx
= ∫ (sec^{2}x + sec x tan x) / (sec x + tan x) dx
Now assume that sec x + tan x = u.
Then (sec x tan x + sec^{2}x) dx = du.
Substituting these values in the above integral,
∫ sec x dx = ∫ du / u = ln u + C
Substituting u = sec x + tan x back here,
∫ sec x dx = ln sec x + tan x + C
Hence proved.
Integral of Sec x by Partial Fractions
To find the integration of sec x by partial fractions, we have to use the fact that sec x is the reciprocal of cos x. Wait! How can this be turned into partial fractions? Let us see.
∫ sec x dx = ∫ 1/(cos x) dx
Multiplying and dividing this by cos x,
∫ sec x dx = ∫ (cos x) / (cos^{2}x) dx
Using one of the trigonometric identities,
∫ sec x dx = ∫ (cos x dx) / (1  sin^{2}x)
Now, assume that sin x = u. Then cos x dx = du. Then the above integral becomes
∫ sec x dx = ∫ du / (1  u^{2})
By partial fractions, 1/(1  u^{2}) = 1/2 [1/(1 + u) + 1/(1  u)]. Then
∫ sec x dx = (1/2) ∫ [ 1/(1 + u) + 1/(1  u) ] du
= (1/2) [ ln 1 + u  ln 1  u ] + C
(We can get this by finding the integrals ∫ [ 1/(1 + u) ] du and ∫ [ 1/(1  u) ] du separately by substitutiong 1 + u = p and 1  u = q respectively).
By a property of logarithms, ln m  ln n = ln (m/n). So
∫ sec x dx = (1/2) ln  (1 + u) / (1  u)  + C
Substituting u = sin x back here,
∫ sec x dx = (1/2) ln  (1 + sin x) / (1  sin x)  + C
Hence proved.
Integral of Sec x by Trigonometric Formulas
We can prove that the integral of sec x to be ln  tan [ (x/2) + (π/4) ]  + C by using trigonometric formulas. We can write sec x as 1/(cos x) where cos x can again be written as sin(x + π/2). So
∫ sec x dx = ∫ 1/(cos x) dx
= ∫ 1/(sin(x + π/2)) dx
Using halfangle formulas, sin A = 2 sin A/2 cos A/2. Applying this,
∫ sec x dx = ∫ 1 / [ 2 sin[ (x/2) + (π/4) ] cos[ (x/2) + (π/4) ] ] dx
= (1/2) ∫ 1 / [ sin[ (x/2) + (π/4) ] cos[ (x/2) + (π/4) ] ] dx
Multiplying and dividing the denominator by cos[ (x/2) + (π/4) ],
∫ sec x dx = (1/2) ∫ 1 / [ sin[ (x/2) + (π/4) ]/cos[ (x/2) + (π/4) ] · cos^{2}[ (x/2) + (π/4) ] ] dx
= (1/2) ∫ sec^{2}[ (x/2) + (π/4) ] / tan[ (x/2) + (π/4) ] dx
Now, assume that tan[(x/2) + (π/4)] = u. From this, (1/2) sec^{2}[ (x/2) + (π/4) ] dx = du.
∫ sec x dx = ∫ du/u = ln u + C
Substituting u = tan[(x/2) + (π/4)] back,
∫ sec x dx = ln  tan [ (x/2) + (π/4) ]  + C
Hence proved.
Integral of Sec x by Hyperbolic Functions
Using hyperbolic functions, we can prove that the integral of sin x to be tanh^{1} (sin x) + C. For this, we use the substitution sec x = cosh t. Now, we have
tan x = √sec²x  1 = √cosh²t  1 = √sinh²t = sinh t
Differentiating both sides,
sec^{2}x dx = cosh t dt
But sec x = cosh t.
(cosh^{2}t) dx = cosh t dt
dx = (cosh t) / (cosh^{2}t) dt = 1/(cosh t) dt
Substituting these values in ∫ sec x dx,
∫ sec x dx = ∫ (cosh t) [1/(cosh t) dt]
= ∫ dt
= t
= cosh^{1}(sec x) + C (This is because sec x = cosh t)
Similarly we can prove that ∫ sec x dx = sinh^{1}(tan x) + C (or) ∫ sec x dx = tanh^{1}(sin x) + C. Can you give them a try?
Therefore, ∫ sec x dx = cosh^{1}(sec x) + C (or) sinh^{1}(tan x) + C (or) tanh^{1}(sin x) + C.
Hence proved.
Important Notes Related to Integration of Sec x:
Here are the formulas of integral of secant x with the respective methods of proving them.
 Using the substitution method,
∫ sec x dx = ln sec x + tan x + C  Using partial fractions,
∫ sec x dx = (1/2) ln  (1 + sin x) / (1  sin x)  + C  Using trigonometric formulas,
∫ sec x dx = ln  tan [ (x/2) + (π/4) ]  + C  Using hyperbolic functions,
∫ sec x dx = cosh^{1}(sec x) + C (or) sinh^{1}(tan x) + C (or) tanh^{1}(sin x) + C
Topics Related to Integral of Sec x:
Here are some topics that you may find helpful while doing the integration of sec x.
Solved Examples on Integration of Sec x

Example 1: Evaluate the definite integral ∫₀^{π/2 }sec x dx if it converges.
Solution:
The integral of sec x is,
∫ sec x dx = ∫ sec x dx = ln sec x + tan x + C
∫₀^{π/2 }sec x dx is obtained by applying the limits 0 and π/2 for this. Then
∫₀^{π/2 }sec x dx = [ ln sec π/2 + tan π/2 + C ]  [ ln sec 0 + tan 0 + C ] = Diverges
This is because sec π/2 = ∞.
Answer: ∫₀^{π/2 }sec x dx diverges.

Example 2: What is the value of ∫ (sec x + tan x) dx?
Solution:
We know that the integrals of secant x and tan x are ln sec x + tan x and ln sec x respectively. Thus,
∫ (sec x + tan x) dx = ∫ sec x dx + ∫ tan x dx
= ln sec x + tan x + ln sec x + C
Answer: ∫ (sec x + tan x) dx = ln sec x + tan x + ln sec x + C.

Example 3: Find the value of ∫ (sec^{2}x + sec x) dx.
Solution:
∫ (sec^{2}x + sec x) dx = ∫ sec^{2}x dx + ∫ sec x dx
We know that ∫ sec^{2}x dx = tan x and ∫ sec x dx = ln sec x + tan x + C. So
∫ (sec^{2}x + sec x) dx = tan x + ln sec x + tan x + C
Answer: ∫ (sec^{2}x + sec x) dx = tan x + ln sec x + tan x + C.
FAQs on Integral of Sec x
What is the Integral of Sec x?
There are multiple formulas for the integral of sec x. But the most popular formula that we use is ∫ sec x dx = ln sec x + tan x + C, where C is the integration constant.
How do You Find the Antiderivative of Sec x?
The antiderivative of sec x is mathematically writen as ∫ sec x dx. Multiply and divide sec x by (sec x + tan x), we get, ∫ sec x dx = ∫ sec x · (sec x + tan x) / (sec x + tan x) dx = ∫ (sec^{2}x + sec x tan x) / (sec x + tan x) dx. By assuming sec x + tan x = u. Then (sec x tan x + sec^{2}x) dx = du. Then the above integral becomes ∫ sec x dx = ∫ du / u = ln u + C = ln sec x + tan x + C.
What is the Integral of Sec x Cos x?
We know that sec x and cos x are reciprocals of each other. So sec x cos x = 1. Thus, ∫ sec x cos x dx = ∫ dx = x + C.
What is the Integration of Sec x Tan x?
We know that the derivative of sec x is sec x tan x. Thus, the integral of sec x tan x is sec x. i.e., ∫ sec x tan x dx = sec x + C.
How to Find the Integral of Sec x by Substitution?
We can write ∫ sec x dx as ∫ sec x dx = ∫ sec x · (sec x + tan x) / (sec x + tan x) dx. Now assume sec x + tan x = u. Differentiating, (sec x tan x + sec^{2}x) dx = du. Then, ∫ sec x dx = ∫ du / u = ln u + C = ln sec x + tan x + C.
What is the Integral of Sec^{2}x Tan x?
We can write ∫ sec^{2}x tan x dx as ∫ sec x (sec x tan x) dx. Assume that sec x = u then sec x tan x dx = du. Then the above integral becomes, ∫ u du = u^{2}/2 + C = (sec^{2}x)/2 + C.
What is the Integral of Sec x from 0 to π/4?
We know that the integral of secant x is ln sec x + tan x. Applying the limits 0 and π/4, we get ∫₀^{π/4 }sec x dx = ln sec π/4 + tan π/4  ln sec 0 + tan 0 = ln √2 + 1  ln 1 = ln √2 + 1.
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