Introduction to integral of cos2t

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 the cos(2t) integral by using different integration techniques.

What is the integral of cos 2t?

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

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

cos = adjacent side/hypotenuse

The cos 2t integral is a common integrand in calculus. It is used to solve different integral problems involving complex functions. Sometimes, we need to evaluate the integrals of a product of two functions, such as the integral of xcos x.

Integration of cos 2t formula

The formula of the integral of cos contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(cos(2t))dt. In mathematical form, the cos2t integration is:

$\int \cos(2t)dt =\frac{\sin(2t)}{2}+ c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. In the integration formula above, replacing the angle 2t by 5x allows us to find the integral of cos(5x).

How to calculate the integral cos2t?

The cos2t integration 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 2t by using:

1. Derivatives
2. Substitution method
3. Definite integral

Integration of cos2t 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 cos2t by using derivatives.

Proof of integral cos2t by using derivatives

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

In derivative, we know that,

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

It means that the derivative of sin x gives us cos(2t). But it has a negative sign. Therefore, to obtain the integral of cosine 2t, we have to use it as the integral of cos, that is:

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

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

$\int \cos(2t)dx = \frac{\sin(2t)}{2} + c$

Using derivatives, we can also integrate cos(4x).

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

Proof of integration of cos 2t by using substitution method

To integrate cos2t by using the substitution method, suppose that:

$y = \cos(2t)$

Differentiating with respect to t,

$\frac{dy}{dt}= -2\sin(2t)$

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

$dy = - 2\sin(2t).dt$

By trigonometric identities, we know that sin(2t) = √1 - cos²2t. Then the above equation becomes,

$dy = -2\sqrt{1 - \cos^2(2t)}.dt$

Now, substituting the value of sin2 2x, such as:

$dy = -2\sqrt{1 – y^2}.dx$

Multiplying both sides by cos(2t),

$\frac{\cos(2t)dy}{-2\sqrt{1 - y^2}} = \cos(2t).dt$

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

$\frac{ydy}{-2\sqrt{1 - y^2}}=\cos(2t).dt$

Integrating on both sides by applying integral,

$\int \frac{y dy}{-2\sqrt{1 -y^2}}=\cos(2t)dt$

Let 1 - y² = u.

Replacing a function with a parameter u to write the integral in simple form, is the method of u-substitution calculator. Then

$-2ydy = du\quad\text{or}\quad ydy = -\frac{1}{2}du$

Then the above left-hand side integral becomes,

$\frac{1}{2}\int \frac{1}{2\sqrt u}du =\int \cos(2t)dt$

$\frac{1}{2}\int \frac{u^{-1/2}}{2}du =\int \cos(2t)dt$

Since the power rule of integration is

$\int x^ndx =\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,

$\frac{1}{2}\times\frac{u^{1/2}}{2(1/2)} + C =\int \cos(2t)dt$

$\frac{u^{1/2}}{2}+ C =\int \cos(2t)dt$

Again substituting u = 1 - y², we get

$\frac{(1 - y^2)^{1/2}}{2}+ C = \int \cos(2t)dt$

And again Substitute y = cos(2t) here,

$\frac{(1 - \cos² 2t)^{1/2}}{2} + C =\int \cos(2t)dt$

$\frac{(\sin² 2t)^{1/2}}{2} + C =\int \cos(2t)dx$

$\frac{\sin(2t)}{2} + C = \int \cos(2t) dt$

Hence the integral of cos2t is sin (2t)/2. Similarly, the integral of cos(2x) is equal to sin(2x)/2.

Integration of cos 2t 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 of cos(2t) by using the definite integral calculator.

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

To compute the integration of cos2t 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(2t) from 0 to π. For this, we can write the integral as:

$\int^\pi_0 \cos(2t)dt = \left|\frac{\sin(2t)}{2}\right|^\pi_0$

Now, substituting the limit in the given function.

$\int^\pi_0 \cos(2t)dt = \frac{\sin(2\pi)}{2} - \frac{\sin(0)}{2}$

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

$\int^\pi_0 \cos(2t)dt = 0$

Which is the calculation of the definite integral of cos(2t). Now to integrate cos2t between the interval 0 to π/4, we just have to replace π by π/4. Therefore,

$\int^{\frac{\pi}{4}}_0 \cos(2t)dt= \left|\frac{\sin (2t)}{2}\right|^{\frac{\pi}{4}}_0$

Now,

$\int^{\frac{\pi}{4}}_0 \cos(2t)dt= \frac{\sin(2π/4)}{2} - \frac{\sin (0)}{2}$

Since sin 0 is equal to 1 and sin π/2 is equal to 1, therefore,

$\int^{\frac{\pi}{4}}_0 \cos(2t)dt = \frac{1}{2} - 0=\frac{1}{2}$

Therefore, the definite integral of cos(2t) is equal to 1/2.