Integral Of Cos^2 (Ax)

Integral of cos^2(ax) along with its formula and proof with examples. Also learn how to calculate integration of cos^2(ax) with step by step examples.

Alan Walker-

Published on 2023-04-13

Introduction to integral of cos^2(ax)

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 calculator 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^2(ax) integral by using different integration techniques.

What is the integral of cos squared ax?

The integral of cos2(ax) is an antiderivative of sine function which is equal to x/2+sin^2(ax)/4a. It is also known as the reverse derivative of sine function which is a trigonometric identity.

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

cos = adjacent side / hypotenuse

Integral of cos2 (ax) formula

The formula of integral of sin contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(cos2(ax))dx. In mathematical form, the integral of cos2(ax) is:

$\int \cos^2(ax)dx = \frac{x}{2} +\frac{\sin^2(ax)}{4a} + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. For a = 2 in the above integration formula, we can calculate the integral of cos^2(2x).

How to calculate the integral of cos2(ax)?

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

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

Integral of cos squared ax 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 integral of two functions combined together. Let’s discuss calculating the integral of cos squared x by using integration by parts.

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

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

$I = \cos(ax).\cos(ax)$

Applying the integral we get,

$I = \int \cos(ax).\cos(ax)dx$

Since the method of integration by parts is:

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

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

$I = \cos(ax).\frac{\sin(ax)}{a} + \int\left[a\sin(ax).\frac{\sin(ax)}{a}\right]dx$

It can be written as:

$I = \cos(ax).\frac{\sin(ax)}{a} + \int[\sin^2(ax)]dx$

Now by using a trigonometric identity sin2(ax) = 1 – cos2(ax)/2. Therefore, substituting the value of cos2x in the above equation, we get:

$I = \cos(ax).\frac{\sin(ax)}{a}+\int\frac{1- \cos2(ax)}{2}dx$

Integrating remaining terms,

$I = \sin(ax).\frac{\cos(ax)}{a} +\frac{x}{2}–\frac{\sin2(ax)}{4a}$





Hence the integral of cos^2x is equal to,


For a = 3, we can calculate the integral of cos^2(3x) by using the above method.

Integral of cos2(ax) 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^2(ax) by using substitution method

To prove theintegral of cos squared (ax) by using u-substitution calculator, suppose that:

$I=\int \cos^2(ax)=\int \left(1- \sin^2(ax)\right)dx$

Further we can sin2(ax) can be substituted as sin2(ax) = 1 – cos2(ax)/2. Then the above equation will become.

$I = x -\int \left(\frac{1-\cos2(ax)}{2}\right)dx$


$I = x – \frac{x}{2} +\frac{\sin2(ax)}{4a}$


$I = \frac{x}{2} +\frac{\sin2(ax)}{4a}$

Hence the integration of cos^2(ax) is verified by using substitution method. To evaluate the non-linear integrand with square root, use our trigonometric substitution calculator.

Integral of cos2(ax) 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 integral cos^2(ax) by using the definite integral.

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

To compute the integration of cos^2(ax) 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 squared (ax) from 0 to 2π.

The definite integral of cos^2(ax) can be written as:

$\int^{2\pi}_0 \cos^2(ax)dx=\left|\frac{x}{2} +\frac{\sin 2(ax)}{4a}\right|^{2\pi}_0$

Substituting the value of limit we get,

$\int^{2\pi}_0 \cos^2(ax)dx=\left[\frac{2π}{2}+\frac{\sin4(aπ)}{4a}\right] - \left[0 +\frac{\sin 0}{4}\right]$

$\int^{2\pi}_0 \cos^2(ax)dx = π - \frac{0}{4}$

Therefore, the integral of cos2 ax from 0 to 2π is

$\int^{2\pi}_0 \cos^2(ax)dx = π$

Which is the calculation of the definite integral of cos^2ax. Now to calculate the integral of sin2 ax between the interval 0 to π, we just have to replace π by π. Therefore,

$\int^\pi_0 \cos^2(ax)dx =\left|\frac{x}{2}+\frac{\sin2(ax)}{4a}\right|^\pi_0$

$\int^\pi_0 \cos^2(ax)dx =\left[\frac{π}{2}+\frac{\sin2(aπ)}{4a}\right] -\left[0 + \frac{sin 0}{4}\right]{2}lt;/p>

$\int^\pi_0 \cos^2(ax)dx=\frac{π}{2}+ 0$

$\int^\pi_0 \cos^2(ax)dx=\frac{π}{2}$

Therefore, the integral of cos2(ax) from 0 to π is π/2.

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