Integral of Sec^3X

Integral of sec cube x along with its formula and proof with examples. Also learn how to calculate integration of sec^3x with step by step examples.

Alan Walker-

Published on 2023-04-20

Introduction to the integral of sec^3x

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 cubic sec x. You will also understand how to compute sec cube integral by using different integration techniques.

What is the integral of sec^3x?

The integral of sec cube x is an antiderivative of secant function which is equal to ½ sec xtan x + (1/2)ln|sec x+tan x| + c. It is also known as the reverse derivative of sec^3x function which is a trigonometric identity. 

The sec cube x function is the reciprocal of the cos function which is written as;

$\sec x = \frac{1}{\cos x}$ 

Integration of sec^3x formula

The formula of integral of sec cubed contains integral sign, coefficient of integration and the function as sec^3x. It is denoted by ∫(sec^3x)dx. In mathematical form, the sec^3x integration is:

 $∫\sec^3xdx = ½(\sec x\tan x) –½ \ln|\sec x+\tan x| + c$

 Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. 

How to calculate the integral of secant cubed?

The integral of sec^3x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of secant by using:

  1. Integration by parts
  2. Substitution method

Integral of sec x cubic 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 integral of sec cubic power x by using integration by parts.

Proof of integration of sec^3x by using integration by parts

Since we know that the function secant sube x can be written as the product of two functions. Therefore, we can calculate the integral of sec^3x by using integration by parts. For this, suppose that:

$I = \sec^3x = \sec x.\sec^2x$ 

Applying the integral we get, 

$I = ∫\sec x.\sec^2xdx$

By using the formula of integration by parts calculator;

$∫[f(x).g(x)]dx = f(x).∫g(x)dx - ∫[f’(x).∫g(x)]dx$

Now replacing f(x) and g(x) by sec x and sec^2x, we get,

$I = \sec x.\tan x - ∫[\sec x\tan x.\tan x]dx$

Where, tan x is the integral of sec^2x.

More simplification, 

$I = \sec x.\tan x - ∫[\tan^2x\sec x]dx$

Now by using a trigonometric identity $tan^2x = 1 + sec^2x$. Therefore, substituting the value of tan2x in the above equation, we get:

$I = \sec x\tan x - ∫\sec x(1 + \sec^2x)dx$

Integrating remaining terms,

$I =\sec x\tan x - ∫\sec xdx– ∫\sec^3xdx$

Since we know that $I = sec^3x$.

$I = \sec x\tan x - \ln|\sec x+\tan x| – I$


$2I = \sec x\tan x - \ln|\sec x+\tan x|$

For more simplification,

$I = ½\sec x\tan x - ½ \ln|\sec x+\tan x|$

Hence the sec^3x integration is equal to,

$∫\sec^3xdx = ½\sec x\tan x - ½ \ln|\sec x+\tan x| + c$

Integral of sec^3x 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 integral sec^3x dx by using the substitution method.

Proof of integral of secant cubed by using substitution method

To integrate sec^3x by using substitution method calculator, suppose that:

$I = ∫sec^3xdx$

Suppose that we can write the above integral as:

$I = ∫\sec x\sec^2xdx$

By using trigonometric identities, we can write the above equation by using sec^2x = 1 – tan^2x, therefore,

$I = ∫\sec x( 1 – \tan^2x)dx$


$I = ∫(\sec x – \sec x\tan^2x)dx$

Now to evaluate first integral, we will use the following steps,

$I_1 = ∫\sec x.dx = \ln|\sec x+\tan x|$

Now to evaluate second integral,

$I_2 = -∫\sec x\tan^2x dx$

Suppose that u = sec x and du = sec xtan x dx, then

$I_2 = -∫u du$

Integrating with respect to u.

$I_2 = -\frac{u^2}{2}$

Substituting the value of u we get,

$I_2 = - \frac{sec^2x}{2}$

Now, using the value of the first and second integral in the above equation to get the final value of the integral.

$I =\ln|\sec x+\tan x| – \frac{\sec^2x}{2} + c$

Hence the integration of sec^3x is verified by using the substitution method. You can also use trigonometric substitution calculator if the integral contains sum or difference of a constant and a variable.


What is the integration of sec 3x?

The calculation of the integral of sec 3x is equal to 1/3 ln|tan 3x+sec 3x| +c. It can be calculated by using substitution method and mathematically, expressed as;

$\int \sec 3xdx=\frac{1}{3} \ln|\tan 3x+\sec 3x|+c$

What is the integral of sec^2x?

The integral fo sec square x is equal to tan x. It is written as;

$\int \sec^2xdx=\tan x+c$

Related Problems

Copyright © 2023