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

What is the integral sine squared 2x?

The integral of sin²(2x) is an antiderivative of sine function which is equal to x/2–sin4x/8. 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

Integration of sin square 2x formula

The formula of integral of sin contains 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:

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

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

How to calculate the integral of sin^2(2x)?

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

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

$I = \sin 2x.\sin 2x$

Applying the integral we get,

$I=∫\sin 2x.\sin 2xdx$

Since the method of integration by parts 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,

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

To eveluate the integral of sin^4x, f(x) and g(x) can be replace by sin^2x in the above formula.

It can be written as:

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

Now by using a trigonometric identity cos²2x = 1+cos4x/2. Therefore, substituting the value of cos²(2x) in the above equation, we get:

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

Integrating remaining terms,

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

Or,

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

Or,

$I=\frac{x}{2}–\frac{\sin4x}{8}$

Hence,

$∫\sin^2(2x)dx=\frac{x}{2}–\frac{\sin4x}{8}$

Integration of sin^2(2x) by using substitution method

The substitution method calculator 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^2(2x) by using substitution method

To proof the integral sine squared 2x by using substitution method, suppose that:

$I=∫\sin^2(2x)=∫(1- \cos^2(2x))dx$

Further we can cos²(2x) can be substituted as cos²(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 sin^2(2x) is verified by using substitution method.

Integral of sin squared (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 of sin square 2x by using the definite integral.

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

To compute the integration of sin^2(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 sin^2 2x from 0 to 2π.

The definite integral of sin^2(2x) can be written as:

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

Substituting the value of limit we get,

$∫^{2π}_0 \sin^2(2x)dx=\left[\frac{2π}{2}-\frac{\sin8π}{8}\right]-\left[0 - \frac{\sin 0}{8}\right]$

$∫^{2π}_0 \sin^2(2x)dx= π – frac{0}{8}$

Therefore, the integral sine squared 2x from 0 to 2π is

$∫^{2π}_0 \sin^2(2x)dx = π$

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

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

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

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

$∫^π_0 \sin^2(2x)dx=\frac{π}{2}$

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