Introduction to integral of cos(4x)
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(4x) integral by using different integration techniques.
What is the integration of cos4x?
The integral of cos(4x) is an antiderivative of sine function which is equal to sin(4x)/4. It is also known as the reverse derivative of sine function which is a trigonometric identity.
The sine function is the ratio of the adjacent side to the hypotenuse of a triangle which is written as:
cos = adjacent side/hypotenuse
Integral of cos4x 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(4x))dx. In mathematical form, the integral of cos(4x) is:
$∫\cos(4x)dx = \frac{\sin(4x)}{4} + c$
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Also, the integral of cos(ln x) can be calculated by replacind cos(4x) with cos(ln x).
How to evaluate integral of cos(4x) with respect to x?
The integral of cos(4x) is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of cosine by using:
- Derivatives
- Substitution method
- Definite integral
Integral of cos 4x 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 antiderivative of cos4x by using derivatives.
Proof of integral of cos(4x) by using derivatives
Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integration of cos4x by using its derivative. For this, we have to look for some derivatives formulas or a formula that gives cos(4x) as the derivative of any function.
In derivative, we know that,
$\frac{d}{dx}(\sin(4x)) = 4\cos(4x)$
It means that the derivative of con(4x) gives us sin(4x). Therefore, to obtain the integral of sine,
$frac{d}{dx}[\sin(4x)] = 4\cos(4x)$
Hence the integral of cos(4x) is equal to sin(4x)/4. It is written as:
$∫\cos(4x)dx = \frac{\sin(4x)}{4} + c$
Thus, this method is also applicable to find the integration of cos(2x).
Integral of cos(4x) 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 4x by using the substitution method.
Proof of integral of cos4x by using substitution method
To prove the integral of cos(4x) by using the substitution method, suppose that:
$y = \cos(4x)$
Differentiating with respect to x,
$\frac{dy}{dx}=-4\sin(4x)$
To calculate integral, we can write the above equation as:
$dy = -4\sin(4x)dx$
By trigonometric identities, we know that sin(4x) = √1 - cos²4x. Then the above equation becomes,
$dy=-4\sqrt{1 - \cos^2(4x)} dx$
Now, substituting the value of cos2 4x, such as:
$dy=-4\sqrt{1 – y^2}.dx$
Multiplying both sides by cos(4x),
$\frac{\cos(4x)dy}{-4\sqrt{1 - y^2}}=\cos(4x)dx$
Again substitute cos(4x) = y on the left side.
$\frac{ydy}{-4\sqrt{1 - y^2}} =\cos(4x)dx$
Integrating on both sides by applying integral,
$∫\frac{ydy}{-4\sqrt{1 - y^2}}=∫\cos(4x) dx$
Now by using u substitution calculator, suppose that 1 - y² = u.
Then
$-2ydy = du\quad\text{or}\quad ydy=\frac{-1}{2}du$
Then the above left-hand side integral becomes,
$\frac{-1}{2}∫\frac{-1}{4\sqrt u} du = ∫\cos(4x)dx$
$\frac{1}{2}∫ \frac{u^{-\frac{1}{2}}}{4}du = ∫\cos(4x)dx$
Since the power rule of integration is
$∫x^n dx= \frac{x^{n+1}}{n+1}+C$
Therefore, by using this formula we get,
$\frac{u^{\frac{1}{2}}}{4}+C=∫\cos(4x) dx$
Again substituting u = 1 - y², we get
$\frac{(1 - y^2)^{\frac{1}{2}}}{4} + C = ∫\cos(4x)dx$
And again Substitute y = cos(4x) here,
$\frac{\left(1-\cos^2(4x)\right)^{\frac{1}{2}}}{4} + C = ∫\cos(4x)dx$
$\frac{\left(\sin²(4x)\right)^{\frac{1}{2}}}{4}+C=∫\cos(4x)dx$
$\frac{\sin(4x)}{4}+C=∫\cos(4x) dx$
Hence the integral of cos4x is sin(4x)/4. Similarly, the integral of cos(5x) is equal to sin(5x)/5.
Integral of cos(4x) 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 cos(4x) by using the definite integral.
Proof of integral of cos(4x) by using definite integral
To compute the antiderivative of cos4x by using a definite integral solver, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of cos(4x) from 0 to π. For this we can write the integral as:
$∫^π_0 \cos(4x)dx=\left|\frac{sin(4x)}{4}\right|^\pi_0$
Now, substituting the limit in the given function.
$∫^π_0 \cos(4x) dx = \frac{\sin(4π)}{4} –\frac{\sin(0)}{4}$
Since sin 0 is equal to 0 and sin π is equal to 0, therefore,
$∫^π_0 \cos(4x) dx = 0$
Which is the calculation of the definite integral of cos(4x). Now to calculate the integration of cos4x between the interval 0 to π/2, we just have to replace π by π/2. Therefore,
$∫^{\frac{π}{2}}_0 \cos(4x)dx=\left|\frac{\sin(4x)}{4}\right|^{\frac{π}{2}}_0$
Now,
$∫^{\frac{π}{2}}_0 \cos(4x)dx=\frac{\sin(4π/2)}{4} –\frac{\sin(0)}{4}$
Since sin 0 is equal to 0 and sin π/2 is equal to 1, therefore,
$∫^{\frac{π}{2}}_0 \cos(4x)dx = 0$
Therefore, the definite integral of cos(4x) is equal to 0.