Introduction to integral of cos^2(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 cos squared 2x. You will also understand how to compute cos square 2x integral by using different integration techniques.

What is the integral of cos^2(2x)?

The integration of cos^2(2x) is an antiderivative of cos function which is equal to x/2 + sin4x/8. It is also known as the reverse derivative of the cosine function, a trigonometric identity.

The sine function is the ratio of adjacent side to the hypotenuse of a triangle which is written as:

cos = adjacent side/hypotenuse

Integral of cos²(2x) formula

The formula of the integral of sin contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(sin²(2x))dx. In mathematical form, the integral of sin^2(2x) is:

$∫\cos^2(2x)dx = \frac{x}{2} + \frac{\sin4x}{8}+ c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Replace cos^2(2x) with cos^2x to evaluate the integral of cos squared.

How to calculate the integral of cos square 2x?

The integral of cos^2(2x) 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:

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

Integral of cos 2x 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 integral of cos squared 2x by using integration by parts.

Proof of cos^2(2x) integral by using integration by parts

Since we know that the function sine squared x can be written as the product of two functions. Therefore, we can calculate the integration of cos^2(2x) by using the integral by parts calculator. For this, suppose that:

$I = \cos 2x.\cos 2x$

Applying the integral we get,

$I = ∫(\cos 2x.\cos 2x)dx$

Since the method of integration by parts is:

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

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

$I = \cos(2x).\frac{sin(2x)}{2} + ∫[\sin(2x).\sin(2x)]dx$

It can be written as:

$I = \cos(2x).\frac{sin(2x)}{2}+ ∫\sin2(2x)dx$

Now by using a trigonometric identity sin²(2x) = 1- cos4x/2. Therefore, substituting the value of sin²(2x) in the above equation, we get:

$I = \cos(2x).\frac{sin(2x)}{2}+ ∫\left(\frac{1- \cos4x}{2}\right)dx$

Integrating remaining terms,

$I = \cos(2x).\frac{sin(2x)}{2}+\frac{x}{2}-\frac{\sin4x}{8}$

Or,

$I = \frac{\sin4x}{4}+\frac{x}{2}-\frac{\sin4x}{8}$

Or,

$I = \frac{x}{2} +\frac{\sin4x}{8}$

Hence the cos^2(2x) integral is equal to,

$∫\cos^2(2x)dx = \frac{x}{2} + \frac{\sin4x}{8}$

Integral of cos^2(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 sin squared by using the substitution method.

Proof of integration of cos^2(2x) by using substitution method

To proof the integral of cos square 2x by using the substitution method, suppose that:

$I =∫\cos^2(2x) = ∫\left(1- \sin^2(2x)\right)dx$

Further we can sin²(2x) can be substituted as sin²(2x) = 1 - cos4x/2. Then the above equation will become.

$I = x - ∫\left(\frac{1- \cos4x}{2}\right)dx$

Integrating,

$I = x –\frac{x}{2} +\frac{\sin4x}{8}$

Moreover,

$I = \frac{x}{2} +\frac{\sin4x}{8}$

Hence the integration of cos²(2x) is verified by using substitution method.

Integral of cos²(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:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integral of sin^2(2x) by using the definite integral.

Proof of integral of cos^2(2x) by using definite integral

To compute the integral of cos square 2x by using a definite integral, we can use the interval from 0 to 2π or 0 to π. Let’s compute the cos^2(2x) integral from 0 to 2π.

The definite integration of cos^2(2x) can be written as:

$∫^{2π}_0 \cos^2(2x) dx = \left|\frac{x}{2} + \frac{\sin4x}{8}\right|^{2π}_0$

Substituting the value of limit we get,

$∫^{2π}_0 cos^2(2x) dx = \left[\frac{2π}{2}+\frac{\sin8π}{8}\right] - \left[0 + \frac{\sin 0}{8}\right]{2}lt;/p>

$∫^{2π}_0 \cos^2(2x)dx = π – 0$

Therefore, the integral of cos2(2x) from 0 to 2π is

$∫^{2π}_0 \cos^2(2x) dx = π$

Which is the calculation of the definite integral of cos^2(2x). Now to calculate the integral of cos between the interval 0 to π, we just have to replace 2π by π. Therefore,

$∫^π_0 \cos^2(2x)dx = \left|\frac{x}{2} +\frac{\sin4x}{8}\right|^π_0$

$∫^π_0 \cos^2(2x)dx = \left[\frac{π}{2}+\frac{\sin4π}{8}\right] - \left[0 + \frac{\sin 0}{8}\right]$

$∫^π_0 \cos^2(2x)dx =\frac{π}{2}- \frac{0}{8}$

$∫^π_0 \cos^2(2x)dx = \frac{π}{2}$

Therefore, the integral of cos2 2x from 0 to π is π/2. Also, learn how to calculate the integral of cos(x^2) by using the definite integral.