Integral Of Sin Square X

Integral of Sin^2x along with its formula and proof with examples. Also learn how to calculate integration of Sin square x with step by step examples.

Alan Walker-

Published on 2023-04-13

Introduction to integral of sin square x

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 integral sine squared by using different integration techniques.

What is the integral of sin^2x?

The integration of sin square x is an antiderivative of sine function which is equal to –cos x. 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 sin2x formula

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

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

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. This formula is also used to calculate the integral of sin^3x.

How to calculate the integral sine squared?

The integral of sin^2x 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 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 integral of two functions combined together. Let’s discuss calculating the integral of sin squared x by using integration by parts.

Proof of integration of sin square x 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^2x by using integration by parts. For this, suppose that:

$I=\sin x.\sin x$

Applying the integral we get,

$I=∫\sin x.\sin xdx$

Since the method of integration by parts calculator 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,

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

It can be written as:

$I=-\sin x.\cos x + ∫\cos^2xdx$

Now by using a trigonometric identity cos2x = 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,

$I=-\sin x\cos x +\frac{1}{2}(x) + \frac{\sin 2x}{4}$


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


$I=\frac{x}{2}–\frac{\sin 2x}{4}$

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

$∫\sin^2xdx = \frac{x}{2} – \frac{\sin 2x}{4}$

Integral of sin squared 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 square x by using substitution method

To proof the integral sine squared by using substitution method calculator, suppose that:

$I = ∫\sin^2x = ∫(1- \cos^2x)dx$

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

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


$I = x –\frac{x}{2}-\frac{\sin2x}{4}$


$I = \frac{x}{2}-\frac{\sin2x}{4}$

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

Integral of sin^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 integration sin square x by using the definite integral. 

Proof of integration of sin square x by using definite integral

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

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

Substituting the value of limit we get,

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

$∫^{2π}_0 \sin^2xdx = π-\frac{0}{4}$

Therefore, the integral of sin2x from 0 to 2π is

$∫^{2π}_0 \sin^2x dx = π$

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

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

$∫^\pi_0 \sin^2xdx=\left[\frac{π}{2}-\frac{\sin π}{4}\right]-\left[0-\frac{\sin 0}{4}\right]$

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

$∫^π_0 \sin^2x dx = \frac{π}{2}$

Therefore, the integral sine squared from 0 to π is π/2. We can also use indefinite integral calculator to evaluate integrals easily. 

Related Problems

Copyright © 2023