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

What is the cos square x integration?

The integral of cos^2x is an antiderivative of the cos square x function which is equal to x/2 +sin 2x/4+c. It is also known as the reverse derivative of the cosine squared function, a trigonometric identity.

The cosine function is the ratio of the opposite side to the hypotenuse of a triangle which is written as:

Sin = opposite side/hypotenuse

Integral of cos²x formula

The formula of the integral of cos square contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(cos2x)dx. In mathematical form, the integral of cos^2x is:

$∫\cos^2xdx=\frac{x}{2}+\frac{\sin2x}{4}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. The above formula is applicable to compute the integral of cos with a higher power. For example, the integral of cos^3x can be calculated by using this formula. 

How to calculate the integral of cos^2x?

The integral of cos^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 sine by using:

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

Integral of cos x 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 x by using integration by parts calculator.

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

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

$I = \cos x.\cos x$

To integrate cos^4x, the above expression can be written as the product to cos^2x twice. 

Applying the integral we get,

$I = ∫(\cos x.\cos x)dx$

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,

$I = \sin x.\cos x + ∫[\sin x.\sin x]dx$

It can be written as:

$I = \sin x.\cos x + ∫\sin^2xdx$

Now by using a trigonometric identity sin²x = 1- cos2x/2. Therefore, substituting the value of cos2x in the above equation, we get:

$I = \sin x.\cos x + ∫\left(\frac{1- \cos2x}{2}\right)dx$

Integrating remaining terms,

$I = \sin x.\cos x+\frac{x}{2}-\frac{\sin 2x}{4}$

Or,

$I = \frac{\sin 2x}{2}+\frac{x}{2}+\\frac{\sin 2x}{4}$

Or,

$I = \frac{x}{2}+\frac{\sin 2x}{4}$

Hence the cos^2x integration by parts is equal to,

$∫\cos^2xdx = \frac{x}{2}+\frac{\sin 2x}{4}+c$

Integral of cos^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 cos squared by using the substitution method.

Proof of Integral of cos^2x by using substitution method

To integrate cos squared x by using the substitution method, suppose that:

$I = ∫\cos^2x = ∫(1- \sin^2x)dx$

Further, we can sin2x can be substituted as sin²x = 1 - cos2x/2. Then the above equation will become.

$I = x - ∫\left(\frac{1- cos2x}{2}\right)dx$

Integrating,

$I = x –\frac{x}{2} +\frac{\sin2x}{4}$

Moreover,

$I =\frac{x}{2}+\frac{\sin2x}{4}+c$

Hence the integration of cos^2x is verified by using the substitution method.

Integral of cos^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 cos^2x by using the definite integral.

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

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

The indefinite integral of cos^2x can be written as:

$∫^{2π}_0 \cos^2x dx =\left|\frac{x}{2}+\frac{\sin 2x}{4}\right|^{2π}_0$

Substituting the value of the limit we get,

$∫^{2π}_0 \cos^2x dx=\left[\frac{2π}{2}+\frac{\sin4π}{4}\right]-\left[0 + \frac{\sin 0}{4}\right]$

$∫^{2π}_0 \cos^2x dx = π - \frac{0}{4}$

Therefore, the integral of cos2x from 0 to 2π is

$∫^{2π}_0 \cos^2x dx = π$

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

$∫^π_0 \cos^2xdx=\left|\frac{x}{2}+\frac{\sin 2x}{4}\right|^{π}_0$

$∫^π_0 \cos^2xdx = \left[\frac{π}{2}+\frac{\sin π}{4}\right]-\left[0 + \frac{\sin 0}{4}\right]$

$∫^π_0 \cos^2xdx = \frac{π}{2}-\frac{0}{4}$

$∫^π_0 \cos^2x dx = \frac{π}{2}$

Therefore, the integral of cos2x from 0 to π is π/2. To evaluate the integral of any function, try our definite integral calculator as it is free to use online.