Integral of Tan x

Integral of tan x along with its formula and proof with examples. Also learn how to calculate integration of tan x with step by step examples.

Alan Walker-

Published on 2023-04-14

Introduction of integral of tan 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 tan. You will also understand how to compute integral of tan by using different integration techniques.

What is the integral of tan?

The integral of tan x is an antiderivative of the tangent function which is equal to ln|sec x|. It is also known as the reverse derivative of tan function which is a trigonometric identity.

The tan function is the ratio of two trigonometric functions sin x and cos x, which is written as:

Tan x= sin x / cos x

The integral of tangent is a common integrand in calculus. It contains a trigonometric function tangent which is used to solve many different integral problems involving tangent functions, such as the integral of tan^2x

Integral of tanx formula

The formula of the integral of sin contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(sin x)dx. In mathematical form, the integration of tanx is:

$\int \tan x dx=\ln|\sec x|+c{2}lt;/p>

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. The above formula can also be used to solve the integral of tan(2x) by using different integration techniques.

How to calculate the integral of tan(x)?

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

  1. Derivatives
  2. Substitution method
  3. Definite integral

Integration of tan x by using u substitution

The u-substitution is a method of integration in calculus. It is used to calculate the integral of a function that is complex to be calculated by usual integration. Let’s discuss calculating the integral tan x by using u-substitution.

Proof of integration of tanx by using u-substitution

Since we know that a function can be replaced by u in the u-substitution calculator. Therefore, we can calculate the integral of tan x by using this technique.

Assume that,

$I = \int(\tan x)dx{2}lt;/p>

By using trigonometric identities, tan x can be written as sin x/cos x. Then,

$I = \int \frac{\sin x}{\cos x}dx{2}lt;/p>

Now, suppose that u = cos x and du = -sin x dx. Therefore,

$I = -\int \frac{1}{u}du{2}lt;/p>


$I = - \ln|u|{2}lt;/p>

Or, it can be written as;

$I = \ln(1) – \ln|u|{2}lt;/p>


$I = \ln \left|\frac{1}{u}\right| + c{2}lt;/p>

Substituting the value of u, we get

$I = \ln|\sec x| + c{2}lt;/p>

Which is the calculation of the integral of tan x. This function tan is also used as a substitution parameter in the trig-substitution calculator.

Integral of tanx by using definite integral

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$\int^b_a f(x)dx = F(b) – F(a)$

Let’s understand the verification of the integration of tan x by using the definite integral.

Proof of integral of tan x by using definite integral

To compute the integration of tanx by using a definite integral, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of tan x from 0 to π. For this, we can write the integral as:

$\int^\pi_0 tan xdx = \ln|\sec x|^\pi_0{2}lt;/p>

Now, substituting the limit in the given function.

$\int^\pi_0 \tan x dx = \ln|\sec (π)| – \ln|sec (0)|{2}lt;/p>

Since cos 0 is equal to 1 and sec π is equal to -1, therefore,

$\int^\pi_0 \tan x dx = \ln(-1) –\ln(1)= \ln(1){2}lt;/p>

Which is the calculation of the definite integration of tan x. Now to calculate the integral of tanx between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$\int^\frac{\pi}{2}_0 \tan x dx = \ln(\sec x)|^\frac{\pi}{2}_0{2}lt;/p>


$\int^\frac{\pi}{2}_0 \tan x dx = \ln\left(\sec \frac{π}{2}\right) – \ln(\sec 0){2}lt;/p>

Since sec 0 is equal to 1 and sec π/2 is equal to 0, therefore,

$\int^\frac{\pi}{2}_0 \tan xdx = 0{2}lt;/p>

Therefore, the definite integral of tanx is equal to 0.

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