Introduction to the integral of tan 2x

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 the integral of tan of 2x by using different integration techniques.

What is the integration of tan 2x?

The integration tan2x is an antiderivative of the tangent function which is equal to (-½ )ln|cos 2x|. It is also known as the reverse derivative of tan 2x function which is a trigonometric identity. 

The tangent function is the ratio of sine to the cosine function which is written as:

Tan = sine/cosine 

Integral of tan2x formula

The formula of the integral tan2x contains the integral sign, coefficient of integration, and the function as tan 2x. It is denoted by ∫(tan 2x)dx. In mathematical form, we can write it as:

 $∫\tan 2xdx = -½ \ln|\cos 2x| + c$

 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 tan(2x)?

The integration of tan2x 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. Substitution method
2. Definite integral

Integral of tan2x 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 tan2x by using u-substitution.

Proof of integral of tan 2x 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 integration of tan2x by using this technique. 

Assume that,

$I = ∫\tan 2xdx$

By using trigonometric identities, tan 2x can be written as sin 2x/cos 2x. Then,

$I = ∫\frac{\sin 2x}{\cos 2x}dx$

Now, suppose that u = cos 2x and du = -2sin 2x dx

Therefore, 

$I = -(½ ) ∫\frac{1}{u}du$ 

Integrating, 

$I = -(½ ) \ln|u|$

Substituting the value of u, we get 

$I =-½ \ln|\cos 2x| + c$

Which is the calculation of the integral of tan 2x.

Integration of tan2x by using definite integral 

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

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integration of tan 2x by using the definite integral.

Proof of integral of tan2x by using definite integral

To compute the integral tan2x by using a definite integral calculator, we can use the interval from 0 to π. Let’s integrate tan 2x from 0 to π. For this, we can write the integral as:

$∫^π_0 \tan 2x dx = -½ \ln|\cos 2x|^π_0$ 

Now, substitute the limit in the given function.

$∫^π_0 \tan 2x dx = -½[ \ln \cos 2(π) – \ln \cos 2(0)]$

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

$∫^π_0 \tan 2x dx = -½ [\ln(1) –\ln(1)]= 0 $

Which is the calculation of the definite integral of tan 2x.