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VOOZH | about |
The integral of sec x is ∫(sec x).dx = ln| sec x + tan x| + C. Integration 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. Learn how to integrate sec x in this article.
In this article, we will understand the formula of the integral of sec x, the Graph of the Integral of sec x, and the Methods of the Integral of sec x.
Table of Content
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.
Formulas for the integral of secant function are:
In these formulas, (C) represents the constant of integration.
Integration of secant x in found using multiple methods that are,
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 + sec x.tan x}.
Step 4: Combine terms and simplify the integrand as much as possible.
Step 5: Evaluate the integral ∫1/u du, 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 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 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
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,
Also, Check
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
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
To find the integral of sec(x), we use a clever technique involving multiplication by a form of one that simplifies the expression. The integral of sec(x) is not straightforward, but we can manipulate it into a more workable form. We multiply and divide the integrand by sec(x)+tan(x), giving us . This transforms the integrand into . Letting u=sec(x)+tan(x), we find its derivative is , which matches the numerator. Thus, the integral becomes, which is . Substituting back, we get ln∣sec(x)+tan(x)∣+C. Therefore, the integral of sec(x) is ln∣sec(x)+tan(x)∣+C.
