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
Also, Check
Integral of Sec x
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, Graph of Integral of sec x, and Methods of Integral of sec x.
Table of Content
- What is Integral of Sec x?
- Integral of Sec x Formula
- Integral of Sec x by Substitution Method
- Integral of Sec x by Partial Method
- Integral of Sec x by Trigonometric Formula
- Integral of Sec x by Hyperbolic Functions
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