Introduction to the Integral of sec x
In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.
Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function sine. You will also understand how to compute secx integration by using different integration techniques.
What is the integral of secx?
The integration of secx is an antiderivative of the cosine function which is equal to ln |sec x+tan x|. It is also known as the reverse derivative of the cosine function which is a trigonometric identity.
The sec function is the reciprocal of the cosine function, that is;
$\sec x= \frac{1}{\cos x}$
The sec x integral is a common integral in calculus, which is used to solve many integral problems such as the integral of sin^2x/cos x.
Secx integration formula
The formula of the integral of sec contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(sec x)dx. In mathematical form, the integral sec x is:
$\int\sec(x)dx=\ln|\sec(x)+\tan(x)|+C$
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral.
How to calculate the integral of secx?
The integral of sec is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of cosine by using:
- Substitution method
- Partial Fraction
- Integration by Parts
Integration of secx by using substitution method
The substitution method involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s understand how to prove the secx integration by using the substitution method.
Proof of Integral of sec by using substitution method
To prove the secx integration formula by using the substitution method, suppose that:
$I = \int \sec(x) dx$
Multiplying and dividing by csc x – cot x, we get
$I = ∫(\sec x\times \frac{\sec x +\tan x}{\sec x + \tan x}dx$
Solving further,
$I = \int \frac{\sec^2(x) + \sec(x)\tan(x)}{\sec(x) + \tan(x)} dx$
Or, it can be written as:
$I = \int \frac{\sec(x)\tan(x) + \sec^2(x)}{\sec(x) + \tan(x)} dx$
Here
$f(x) = \sec(x) + \tan(x)$
and it’s derivative as
$f'(x) = \sec(x)\tan(x) + \sec^2(x){2}$
therefore, the above integral can be written as;
$I = \int \frac{f'(x)}{f(x)} dx{2}$
Integrating,
$I = \ln|f(x)| + c$
Using the value of f(x),
$I = \ln|\sec(x) + \tan(x)| + C$
Hence we have verified the integration of secx. You can also use our trigonometric calculator to find integration of a function by using trigonometric substitution.
Integral of secant x by using partial fraction
Partial fraction is used to decompose rational expressions. It is also used to find the integral of any rational function easily. Therefore we can use this method to calculate the integration of sec x.
Proof of sec x integral by using partial fraction
To proof the sec x integration,
$∫\sec xdx = ∫\frac{1}{\cos x}dx$
Multiplying and dividing this by sin x,
$∫ \sec x dx = ∫\frac{\cos x}{\cos^2x}dx$
Using one of the trigonometric formulas,
$∫\sec x dx = ∫\frac{\cos x}{1 - \sin^2x} dx$
Now, assume that sin x = u. Then cos x dx = du. Then the above integral becomes
$∫\sec x dx = ∫ \frac{du}{1 - u^2}$
Now by partial fraction calculator,
$\frac{1}{1 - u^2} = \frac{1}{2}\left[\frac{1}{1+ u} + \frac{1}{1 - u}\right]$
Then,
$∫\sec x dx =\frac{1}{2}∫\left[\frac{1}{1+ u} - \frac{1}{1 - u}\right]du$
$∫\sec x dx = \frac{1}{2}[ \ln|1+ u| - \ln|1 - u|] + C$
By using logarithmic laws,
$∫\sec x dx = \frac{1}{2} \ln \left|\frac{1+u}{1- u}\right| + C$
Now substituting the value of u,
$∫\sec x dx = \frac{1}{2} \ln \left|\frac{1+\sin x}{1- \sin x}\right|+ C$
Hence we have verified the integral of secx by using the partial fraction.
Secx integration by using integration by parts
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. Let’s discuss calculating the integration of secx by using integration by parts.
Secx integral proof by using integration by parts
Since we know that the function sec x can be written as the product of two functions. Therefore, we can integrate it by using integration by parts. For this, suppose that:
$u = \sec x$
Taking the derivatives, we get:
$\frac{du}{dx} = \sec x\tan x$
Plugging these values into the integration by parts formula, we get:
$∫\sec xdx = x\sec x - ∫x \sec x\tan xdx$
Now, we can simplify the second term using substitution. Let z = sin(x), then dz/dx = cos(x) and dx = dz/cos(x). Substituting these values, we get:
$∫\sec xdx = x\sec(x) - ∫\frac{z}{1-z^2}dz$
Now, we can use partial fraction decomposition to simplify the second term:
$∫\frac{z}{1-z^2} dz = \frac{1}{2} ln\left|\frac{1+z}{1-z}\right| + C$
where C is the constant of integration.
Putting it all together, we get:
$∫\sec x dx = x\sec x - \frac{1}{2} \ln\left|\frac{1+\sin x}{1-\sin x}\right| + C$
where C is the constant of integration.
