Integral of Sec x

Integral of the secant function, denoted as ∫(sec x).dx and is given by: ∫(sec x).dx = ln| sec(x) + tan(x)| + C. Sec x is one of the fundamental functions of trigonometry and is the reciprocal function of Cos x.

In this article, we will understand the formula of the integral of sec x, Graph of Integral of sec x, and Methods of Integral of sec x.

What is Integral of Sec x?

Integral of the secant function, denoted as ∫(sec x).dx represents the area under the curve of secant from a given starting point to a specific endpoint along the x-axis. Mathematically, the integral of secant function is commonly expressed as

∫(sec x).dx = ln| sec(x) + tan(x)| + C

where (C) represents the constant of integration. This integral often arises in calculus problems involving trigonometric functions and has various applications in fields such as physics, engineering, and mathematics.

Integral of Sec x Formula

Formulas for the integral of secant function are:

• ∫(sec x).dx = ln |sec(x) + tan(x)| + C
• ∫(sec x).dx = 1/2ln |(1 + sin x)/(1 – sin x)| + C

In these formulas, (C) represents the constant of integration.

Integration of secant x in found using multiple methods tht are,

• By using Substitution Method
• By using Partial Fractions
• By using Trigonometric Formulas
• By using Hyperbolic Functions

Integral of Sec x by Substitution Method

Integral of Sec x by substitution method is found by the steps added below,

Step 1: Choose an appropriate substitution to simplify the integral. In this case, a common choice is u = tan(x) + sec(x).

Step 2: Calculate the differential of (u) with respect to (x), denoted as (du), using the chain rule. For the chosen substitution, du = sec²(x) + sec(x) tan(x), dx

Step 3: Rewrite the integral in terms of the variable (u). The integrand becomes (1/u) and (dx) is replaced by du/{sec²x + secx.tan x}.

Step 4: Combine terms and simplify the integrand as much as possible.

Step 5: Evaluate the integral ∫1/udu, which yields (ln|u| + C), where (C) is the constant of integration.

Step 6: Replace (u) with the original expression involving (x). The result is (ln| tan(x) + sec(x)| + C), where C represents the constant of integration.

Thus,

∫sec (x)dx = A.ln|sec x + tan x| – B.ln|cosec x + cot x| + C

where,

• A and B are constants Determined from Partial Fraction Decomposition
• C is Constant of Integration

Integral of Sec x by Partial Method

Integral of secant function ∫(sec x).dx, can be evaluated using the partial fraction decomposition method with the following steps:

Step 1: Rewrite sec(x) as 1/cos(x)

Step 2: Express 1/cos(x) as (A/cos(x) + B/sin(x)

Step 3: Multiply both sides by cos(x) to eliminate the denominator and then separately set (x = 0) and (x = π/2) to solve for (A) and (B).

Step 4: Rewrite (∫sec(x), dx as ∫Acos(x) + Bsin(x) dx.

Step 5: Integrate Acos(x) and Bsin(x) separately. This yields (A ln| sec(x) + tan(x)|) and (-B ln| csc(x) + cot(x)|) respectively.

Step 6: Combine the two integrals to get the final result.

Here, integral of secant function using the partial fraction decomposition method:

∫sec (x)dx = A.ln|sec x + tan x| – B.ln|cosec x + cot x| + C

where,

• A and B are constants Determined from Partial Fraction Decomposition
• C is Constant of Integration

Integral of Sec x by Trigonometric Formula

Integral of the secant function, (∫sec(x) , dx), can be evaluated using trigonometric formulas. One common approach involves using the identity sec(x) = 1/cos(x) and then integrating 1/cos(x).

Step 1: Rewrite sec(x) as ( 1/cos(x)).

Step 2: Replace sec(x) with (1/cos(x)) in the integral

Step 3: Integrate (1/cos(x)) with respect to (x). This yields ln |sec x + tan x| + C, where (C) is the constant of integration.

So, integral of secant function using the trigonometric formula is:

∫ sec x dx = ln |sec x + tan x| + c

where, C is Constant of Integration

Integral of Sec x by Hyperbolic Functions

Hyperbolic functions can also be used to find integral of sec x. We know that,

tan x = √(sec²x) – 1…(i)

tan x = √(cosh²t) – 1…(ii)

tan x = √(sinh²t) = sinh t…(iii)

From eq. (iii)

tan x = sinh t

Differentiating both sides,

sec²x dx = cosh t dt

Also, sec x = cosh t

(cosh²t) dx = cosh t dt

dx = (cosh t) / (cosh²t) 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

Thus,

∫sec x dx = cosh^-1(sec x) + C

Also, ∫sec x dx can also be found as,

• ∫sec x dx = sinh^-1(sec x) + C
• ∫sec x dx = tanh^-1(sec x) + C

Examples on Integral of Sec x

Various examples on Integral of Sec x

Example 1. Evaluate ∫sec(x).dx

Solution:

sec(x) = 1/cos(x)

Substitute u = sin(x), so du = cos(x)dx.

Now, (∫cos(x). dx = ∫1/u.du)

= ∫1/u.du

= ln |u| + c

= ln |sin (x)| + c

Example 2. Determine ∫sec(x).tan(x).dx

Solution:

Let,

• u = sec(x)
• du = sec(x) tan(x) dx

Thus,

= ∫sec(x) tan(x), dx

= ∫du

= u + C

= sec(x) + C

Example 3. Find ∫sec²(x).dx.

Solution:

= ∫sec²(x).dx

Using Power Rule for Integration

= tan(x) + C

So, ∫sec²(x), dx = tan(x) + C, where C is Constant of Integration

Example 4. Calculate ∫sec(x)/tan(x).dx.

Solution:

Let,

• u = tan(x)
• du = sec²(x).dx

Substituting (u) and (du), we get:

= ∫ 1/u.du

= ln|u| + C

Substituting, u = tan(x)

= ln| tan(x)| + C

Practice Questions on Integral of Sec x

Some questions related to Integral of Sec x are

Q1: Evaluate ∫secx.tan²x dx

Q2: Determine ∫secx.cotx dx

Q3: Find ∫4.secx.tanx dx

Q4: Calculate ∫secx.cosxdx

Q5: Solve ∫sec (x)dx

FAQs on Integral of Sec x

What is Integral of Sec x?

Integral of secant function, denoted as ∫sec(x)dx, is commonly expressed as (ln |sec(x) + tan(x)| + C), where (C) represents constant of integration.

How to Calculate Integral of Secant?

Integral of secant function, is found using various methods that are added in the article above.

What is Integral of Sec x Cos x?

Integral of Sec x Cos x is, ∫ sec x cos x dx = ∫ 1.dx = x + C

What is Integral of sec x tan x?

Formula for integration of sec x.tan x is ∫(sec x.tan x)dx = sec x + C