Integral of Sec x

The reciprocal function of sec x is cos x, since sec x = 1 / cos x. The integral function is denoted by the symbol ∫. So integral sec x is ∫sec x dx. One of the popular formulas to find the integral of sec x is ∫sec x dx = In |sec x + tan x| + C, where C is the integration constant, and In is the natural logarithm. 

There are multiple ways to find the integration. In this section, we will discuss some common methods we use to find the integral of sec x.

• Substitution method
   
• Partial method
   
• Trigonometric formula
   
• Hyperbolic function

When the given function is complex or direct integration is not possible, we use the substitution method. Here we use a new variable to substitute. Let’s find the integral of sec x using the Substitution method.

Multiplying and dividing by sec x + tan x 

That is, ∫sec (x) dx = ∫sec(x). (sec(x) + tan(x)) / sec(x) + tan (x) 

Expanding the numerator

sec(x). (sec(x) + tan(x)) = sec² (x) + sec (x) tan (x) 

So, ∫sec (x) dx = ∫sec² (x) + sec (x) tan (x) / sec (x) + tan (x) dx

Let u = sec (x) + tan (x)

Differentiate u with x: du/dx = sec (x) tan (x) + sec²(x)

Therefore, du = (sec (x) tan (x) + sec²(x)) dx

Hence, the numerator is equal to the du.

Substituting, u = sec (x) + tan (x); du = (sec (x) tan (x) + sec²(x)) dx

∫sec² (x) + sec (x) tan (x) / sec (x) + tan (x) dx = ∫du/u = In |u| + C

Here, u = sec (x) + tan (x)

So, In |u| + C = In |sec (x) + tan (x)| + C

Therefore, ∫sec (x) dx = In |sec (x) + tan (x)| + C

In this method, an improper-looking rational function is broken down into a proper rational function. Sec(x) = 1/cos(x)

∫sec (x) dx =  ∫1 / cos(x)

Multiplying and diving by cos(x)

 ∫sec (x) dx =  ∫cos(x) / (cos²x) dx

In trigonometry identities, cos²x = 1 - sin²x

So,  ∫sec (x) dx =  ∫cos(x) / (1 - sin²x) dx

u = sin(x), du = cos(x) dx, substituting the value in

∫cos(x) / (1 - sin²x) can be written as ∫du / (1 - u)²

So, ∫sec (x) dx =  ∫du / (1 - u)²

Using partial fraction decomposition on 1 / (1 - u²)

That is, 1 / 1 - u² = A / (1 + u) + B / (1 - u) 

1 = A(1 - u) +B(1 +u)

1 = A - Au + B + Bu

1 = (A + B) + (-A + B)u

That is A + B = 1

-A + B = 0 → B - A = 0 → B = A

Substituting, B = A into A + B = 1

A + A = 1 →  A = ½, and since B = A, B = ½

Therefore, 1 / 1 - u² = (1/2)/1 + u + (1/2)/1 - u 

 ∫1 / 1 - u² du = ½  ∫1 / 1 + u du + ½  ∫ 1 / 1 - u du

 ∫1 / 1 + u = In| 1 + u| and  ∫1 / 1 - u = -In |1 - u| 

So,  ∫1 / 1 - u² du = ½  In| 1 + u|  + ½  In| 1 - u| + C
∫1 / 1 - u² du = ½ In |1 + u/1 - u| + C

As u = sin(x), so substituting u = sin(x)

∫ sec(x) dx = ½ In |1 + sin(x)/1 - sin(x)| + C

Therefore, ∫ sec(x) dx = ½ In | 1 + sin(x) / 1 - sin(x) | + C

Trigonometric formulas use trigonometric identities to find the value of sec x. Sec x is equal to 1 / cos x.

∫ sec(x) = ∫ 1 / cos(x) dx

In trigonometric identity, cos(x) = sin(x +π/2)

Thus, ∫ sec(x) = ∫ 1 / sin(x +π/2) dx

Rewriting sine function using half-angle formula

sin(A) = 2sin(A/2) cos(A/2)

Substituting it in sin(x +π/2)

sin(x +π/2) = 2sin(x/2 +π/4) cos(x/2 +π/4)

Finding the ∫ sec(x) 

That is ∫ sec(x) dx = ∫ 1/ 2sin(x/2 +π/4) cos(x/2 +π/4) dx

Factor out ½
That is, ∫ sec(x) dx = ½ ∫ 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 = ½ ∫ 1/ sin(x/2 +π/4) / cos(x/2 +π/4). cos²((x/2) + (π/4)) dx
= ½ ∫ sec²((x/2) + (π/4)) / tan ((x/2) +(π/4)) dx

Considering u = tan((x/2) +(π/4)) 

Derivate of tan(A), d/dx [tan(A)] = sec² (A) dA/dx

Differentiate u = d/dx tan (x/2 + π/4) = ½ sec² (x/2 + π/4)
du = ½ sec²(x/2 + π/4) dx 

∫ sec(x) dx = ∫ 1/u du

Integral of 1/u is ∫ 1/u du = In|u| + C

Here, u = tan (x/2 + π/4)

So, ∫ sec(x) dx = In | tan (x/2 + π/4) | + C

The hyperbolic function is the same as a trigonometric function for circles. Here sinh, cosh, tanh, coth, sech, and csch are the functions. Let's find the value of ∫ sec(x) dx.

In trigonometric identities, tan(x) = √sec²(x) - 1

In hyperbolic identity, cosh²(t) - sinh²(t) = 1

That is tanh²(t) = cosh²(t) - 1

So, tan(x) = sinh(t)

Differentiating both sides

 sec²x dx = cosh t dt

sec x = cosh t

cosh²t dx = cosh t dt

dx = (cosh t) / cosh²(t) dt

= 1 / cosh t dx

Substituting the ∫ sec x dx

= ∫ sec x dx

= ∫ (cosh t) (1,(cosh t) dt)

= ∫ dt

= t

= cosh^-1(sec x) + C

So, ∫ sec x dx = cosh^-1(sec x) + C

Tips and tricks make it interesting for kids to learn integration. To master integration, kids can use these tips and tricks.

• Memorizing the integrals: By memorizing the equations, students can apply and use it when finding a number's integral.

• Following the correct method: There are different ways to find the integrals, so we should use the correct method to get the correct answer.  

• Memorize the trigonometric identities: By identifying the correct trigonometric identities students can easily apply them when finding the integrals. 

• Trigonometric identities: The equations related to trigonometric function. For example, sec²(x) - tan²(x) = 1

• Integration constant (C): Integration constant is a number which could be added when we integrate a function 

• Substitution Method: It is a method used to find the value of integration. It is used when the function is complex or director integration is not applicable; we use a substitution method.