Introduction to integral of tan²x

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 tangent squared. You will also understand how to compute the integration of tan square x by using different integration techniques.

What is the integral of tan²x?

The integral of tan^2x is an antiderivative of tan^2x function which is equal to tan x– x + c. It is also known as the reverse derivative of the tan^2x function which is a trigonometric identity.

The tan function is the ratio of the sine and cosine function which is written as:

tan = sine/cosine

The integral of tan square x is a common integral expression in calculus. It involves the square of a trigonometric function tan which is used to solve different integral expressions.

Integral of tan²x formula

The formula of the tan square x integration contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(tan²x)dx. In mathematical form, the antiderivative of tan^2x is:

$\int \tan^2xdx = \tan x – x +c$

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

How to calculate the integral of tan^2x?

The integral of tangent squared is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the antiderivative of tan^2(x) by using:

1. Integration by parts
2. Substitution method
3. Definite integral

Integral of tan²x squared 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 tan^2x by using integration by parts.

Proof of tan square x integration by using integration by parts

Since we know that the function tan squared x can be written as the product of two functions. Therefore, we can calculate the integration of tan square x by using integration by parts. For this, suppose that:

$I = \frac{\sin^2x}{\cos^2x}$

Or,

$I = \sin x.\frac {\sin x}{cos^2x}$

Applying the integral we get,

$I = \int \sin x.\frac{\sin x}{\cos^2x}dx = \int \sin x.(\tan x.\sec x)dx$

Since the method of integration by parts formula is:

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

Now replacing f(x) and g(x) by sin x, we get,

$I = \sin x.\int \tan x.\sec xdx – \int \left[\frac{d}{dx}(\sin x).\int \tan x.\sec x.dx\right]dx$

It can be written as:

$I = \sin x.\sec x –\int \cos x.\sec x.dx$

Now by using a trigonometric identity cos x.sec x = 1. Therefore, substituting the value of cos x.sec x in the above equation, we get:

$I = \sin x.\frac{1}{cos x} –\int dx$

Integrating remaining terms,

$I = \tan x – x$

Hence the antiderivative of tan^2x is equal to,

$\int \tan^2xdx = \tan x – x$

Integration of tan^2x 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 of tan squared by using the substitution method.

Proof of integration of tan square x by using substitution method

To prove the integral of tan^2x by using the substitution method, suppose that:

Further, we can cos²x can be substituted as sin²x = 1 – cos²x. Then the above equation will become.

Again,

Moreover,

$I = \int (\sec^2x – 1)dx$

Now integrating,

$I = \tan^2x – x + c$

Hence the integration of tan²x is verified by using the u-substitution calculator.

Integral of tan^2x 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:

$\int^b_af(x) dx = F(b) – F(a)$

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

Proof of integral of tangent squared by using definite integral

To integrate tan^2x by using a definite integral, we can use the interval from 0 to π/4 or -π/4 to 0. Let’s compute the tan square x integration from 0 to 2π. The definite integral of tan^2x can be written as:

$\int^\frac{\pi}{4}_0 \tan^2x dx = [\tan x – x]|^\frac{\pi}{4}_0$

Substituting the value of the limit we get,

$\int^\frac{\pi}{4}_0 \tan^2x dx = [\tan \frac{\pi}{4} – \frac{\pi}{4}] - [\tan 0 – 0]$

$\int^\frac{\pi}{4}_0 \tan^2x dx = 1 – \frac{\pi}{4}$

Therefore, the integral of tan2x from 0 to π/4 is

$\int^\frac{\pi}{4}_0 \tan^2x dx= 1 – \frac{\pi}{4}$

Which is the calculation of the definite integral of tan^2x. Now calculate the integration of tan square x between the interval from –π/4 to 0. Therefore,

$\int^0_{-\frac{\pi}{4}} \tan^2x dx = [\tan x – x] |^0_{-\frac{\pi}{4}}$

$\int^0_{-\frac{\pi}{4}} \tan^2x dx = [\tan 0 – 0] - [\tan (\frac{-\pi}{4}) + \frac {\pi}{4}]$

$\int^0_{-\frac{\pi}{4}} \tan^2x dx = 1 – \frac{\pi}{4}$

Therefore, the antiderivative of tan^2x from –π/4 to 0 is 1 – π/4. If an integrand is not bounded between two points, you can use our indefinite integral calculator to solve such integrals.