Integral Of Sin^2(Ax)

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

Alan Walker-

Published on 2023-04-13

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

What is the integral of sin^2(ax)?

The integral of sin^2ax is an antiderivative of the sine function which is equal to x/2 - (sin2ax)/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:

Sin = opposite side / hypotenuse

Integral of sin2(ax) formula

The formula of integral of sin contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(sin2ax)dx. In mathematical form, the integral sin^2(ax) is:


Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.

How to calculate the sin^2(ax) integral?

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

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

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

Proof of integrate sine squared ax 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 integral of sin^2(ax) by using integration by parts. For this, suppose that:

$I = \sin ax.\sin ax$

Applying the integral we get,

$I = ∫\sin ax.\sin axdx$

Since the integration by parts formula calculator is:

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

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

$I=-\sin ax.\frac{\cos ax}{a}+∫[\cos ax.\cos ax]dx$

It can be written as:

$I=-\sin ax.\frac{\cos ax}{a}+∫\cos^2axdx$

Now by using a trigonometric identity cos2 ax = 1+cos2ax/2. Therefore, substituting the value of cos2ax in the above equation, we get:

$I=-\sin ax\frac{\cos ax}{a}+∫\left(\frac{1+\cos2ax}{2}\right)dx$

Integrating remaining terms,

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

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


$I=\frac{x}{2}–\frac{\sin 2ax}{4a}$

Hence the sin^2(ax) integral is equal to,

$∫\sin^2(ax)dx =\frac{x}{2}–\frac{\sin 2ax}{4a}$

Integration of sin^2ax 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 Integral of sin^2ax by using substitution method

To proof the integral sin^2ax by using substitution method, suppose that:

$I=∫\sin^2ax = ∫(1- \cos^2ax)dx$

Further we can cos2 ax can be substituted as cos2ax = 1+cos2ax/2. Then the above equation will become.

$I = x - ∫\left(\frac{1+ \cos2ax}{2}\right)dx$


$I = x – \frac{x}{2}-\frac{\sin2ax}{4}$


$I = \frac{x}{2}-\frac{\sin2ax}{4a}$

Hence the integration of sin^2ax is verified by using substitution method. You can also use our u substitution calculator to verify the above calculations.

Integral of sin^2ax 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 formula used by the definite integral calculator can be written as:

$∫^b_a f(x) dx = F(b) – F(a)$

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

Proof of integral sine squared ax by using definite integral

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

The indefinite integral of sin^2ax can be written as:

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

Substituting the value of limit we get,

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

$∫^{2π}_0 \sin^2ax dx = π - \frac{0}{4}$

Therefore, the integral of sin2ax from 0 to 2π is

$∫^{2π}_0 \sin^2ax dx = π$

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

$∫^π_0 \sin^2ax dx=\left|\frac{x}{2}-\frac{\sin 2ax}{4}\right|^π_0$

$∫^π_0 \sin^2axdx=\left[\frac{π}{2}-\frac{\sin aπ}{4a}\right]-\left[0 -\frac{\sin 0}{4a}\right]$

$∫^π_0 \sin^2ax dx=\frac{π}{2}-\frac{0}{4a}$

$∫^π_0 \sin^2ax dx = \frac{π}{2}$

Therefore, the integral of sin2 ax from 0 to π is π/2.


What is the integral of sin 2ax?

Integral of sin^2ax is the calculation of integral of sin squared x. It is written as ∫sin2ax dx and is equal to x/2 - sin2ax/4a.

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