Introduction to the 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. You will also understand how to compute cos(2x) integral by using different integration techniques.

What is the integral of cos2x?

The integral of cos(2x) is an antiderivative of the cos2x function which is equal to sin(2x)/2. It is also known as the reverse derivative of the cosine function which is a trigonometric identity.

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

cos = adjacent side/hypotenuse

Integration of cos2x formula

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

$∫\cos(2x)dx = \frac{\sin 2x}{2}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. In the above formula, cos 2x can be replaced with cos 4x to calculate the integral cos(4x).

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 cosine by using:

1. Derivatives
2. Substitution method
3. Definite integral

Integral of cos(2x) by using derivatives

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. Therefore, we can use the derivative to calculate the integral of a function. Let’s discuss calculating the integral of cos(2x) by using derivatives.

Proof of cos2x integration by using derivatives

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of cos2x by using its derivative. For this, we have to look for some derivative formulas or a formula that gives cos(2x) as the derivative of any function.

In derivative, we know that,

$\frac{d}{dx}(\sin 2x) = 2\cos(2x)$

It means that the derivative of sin(2x) gives us cos(2x). But it has a negative sign. Therefore, to obtain the integral of sine, we have to multiply the above equation by the negative sign, that is:

$\frac{d}{dx} (\sin 2x) = 2\cos(2x)$

Hence the integral of cos(2x) is equal to the sin(2x)/2. It is written as:

$∫\cos(2x)dx = \frac{\sin 2x}{2}+c$

Integral of cos(2x) by using the 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 (2x) by using the substitution method.

Proof of integration of cos 2x by using the substitution method

To integrate cos 2x by using the substitution method, suppose that:

$y = \cos(2x)$

Differentiating with respect to x,

$\frac{dy}{dx}= -2\sin(2x)$

To calculate integral, we can write the above equation as:

$dy = -2\sin(2x)dx$

By trigonometric identities, we know that sin 2x = √1 - 2sin²x. Then the above equation becomes,

$dy = -\sqrt{1 - 2\sin^2x}.dx$

Now, substituting the value of sin2 x, such as:

$dy = -\sqrt{1 – 2y^2}.dx$

Multiplying both sides by cos(2x),

$-\frac{\cos(2x)dy}{\sqrt{1 - 2y^2}}=\cos(2x)dx$

Again substitute cos(2x) = y on the left side.

$-\frac{ydy}{\sqrt{1 - 2y^2}}=\cos(2x)dx$

Integrating on both sides by applying integral,

$-∫\frac{ydy}{\sqrt{1 - 2y^2}}=∫\cos(2x)dx$

Let 1 - 2y² = u.

Then

$-4y dy = du\quad \text{or}\quad ydy=-\frac{1}{4}du$

Then the above left-hand side integral becomes,

$\frac{1}{4}∫\frac{1}{\sqrt u}du=∫\cos(2x)dx$

$\frac{1}{4}∫u^{-\frac{1}{2}}du=∫\cos(2x)dx$

Since the power rule of integration is

$∫x^n dx=\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,

$\frac{1}{4}\left(\frac{u^{1/2}}{1/2}\right)+C =∫\cos(2x) dx$

$\frac{u^{\frac{1}{2}}}{2}+C=∫\cos(2x)dx$

Again substituting u = 1 - y², we get

$\frac{(1 - y²)^{\frac{1}{2}}}{2}+C=∫\cos(2x)dx$

Replacing an integrand or a part of integrad with a parameter u is known as u-substitution method. And again Substitute y = cos(2x) here,

$\frac{(1 - \cos^22x)^{\frac{1}{2}}}{2} + C=∫\cos(2x)dx$

$\frac{(\sin^22x)^{\frac{1}{2}}}{2} + C =∫\cos(2x) dx$

$\frac{\sin 2x}{2}+C=∫\cos(2x)dx$

Hence the integral of cos(2x) is sin 2x/2.

Integration of cos2x 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 cos2x by using the definite integral.

Proof of integral of cos(2x) by using definite integral

The definite integral is usually used to calculate the area under a curve. To compute the integral of cos(2x) by using a definite integral, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of cos(2x) from 0 to π. For this, we can write the integral as:

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

Now, substitute the limit in the given function.

$∫^π_0 \cos(2x)dx = \frac{\sin2(π)}{2}-\frac{\sin2(0)}{2}$

Since sin 0 is equal to 0 and sin π is equal to 0, therefore,

$∫^π_0 \cos(2x)dx = 0$

Which is the calculation of the definite integral of cos(2x). Now to calculate the integration of cos2x between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

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

Now,

$∫^{\frac{π}{2}}_0 \cos(2x)dx=\frac{\sin(π)}{2} - \frac{\sin2(0)}{2}$

Since sin 0 is equal to 0 and cos π/2 is equal to 0, therefore,

$∫^{\frac{π}{2}}_0 \cos(2x)dx=0$

Therefore, the definite integral of cos(2x) is equal to 0.