Introduction to integral of cos(pix)

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 calculator. 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(πx) integral by using different integration techniques.

What is the integral of cos pi x?

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

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

cos = adjacent side / hypotenuse

Integral of cos(πx) formula

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

$∫\cos(πx)dx =\frac{\sin(πx)}{π} + c{2}lt;/p>

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. Also, the integral of cos(lnx) can be calculated by replacind cos(πx) with cos(ln x).

How to calculate the integral of cos(πx)?

The integral of cos(πx) 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:

1. Derivatives
2. Substitution method
3. Definite integral

Integral of cos(πx) 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 function by using integral calculator. Let’s discuss calculating the antiderivative of cos pi x by using derivatives.

Proof of integral of cos pi x by using derivatives

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

In derivative, we know that,

$\frac{d}{dx}(\sin πx) = π\cos(πx){2}lt;/p>

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

$\frac{d}{dx}(\sin πx) = π\cos(πx){2}lt;/p>

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

$∫\cos(πx)dx = \frac{\sin(πx)}{π} + c{2}lt;/p>

Thus, the integral of cos(e^x) can be calculated by using derivatives.

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

Proof of antiderivative of cos pi x by using substitution method

To proof the integral of cos(πx) by using the u substitution method, suppose that:

$y = \cos(πx){2}lt;/p>

Differentiating with respect to x,

$\frac{dy}{dx} = - π\sin(πx){2}lt;/p>

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

$dy = -π\sin(πx).dx{2}lt;/p>

By trigonometric identities, we know that sin(πx) = √1 - cos²πx. Then the above equation becomes,

$dy = -π\sqrt{1 - \cos^2(πx)}.dx{2}lt;/p>

Now, substituting the value of cos2πx, such as:

$dy = -π\sqrt{1 – y^2}.dx{2}lt;/p>

Multiplying both sides by cos(πx),

$-\frac{\cos(πx)dy}{π\sqrt{1 - y^2}} = \cos(πx).dx{2}lt;/p>

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

$-\frac{ydy}{π\sqrt{1 - y^2}} = \cos(πx).dx{2}lt;/p>

Integrating on both sides by applying integral,

$-∫\frac{y dy}{π\sqrt{1 - y^2}} = ∫\cos(πx)dx{2}lt;/p>

Let 1 - y² = u.

Then

$-2y dy = du\quad \text{or}\quad ydy=\frac{-1}{2}du.{2}lt;/p>

Then the above left-hand side integral becomes,

$\frac{-1}{2}∫\frac{-1}{π\sqrt u}du=∫ \cos(πx)dx{2}lt;/p>

$\frac{1}{2π}∫u^{\frac{-1}{2}}du=∫ \cos(πx)dx{2}lt;/p>

Since the power rule of integration is

$∫x^n dx = \frac{x^{n+1}}{n+1}+C{2}lt;/p>

Therefore, by using this formula we get,

$\frac{1}{2π}\left(\frac{u^{\frac{1}{2}}}{\frac{1}{2}}\right)+C =∫\cos(πx)dx{2}lt;/p>

$\frac{u^{\frac{1}{2}}}{π} + C =∫\cos(πx)dx{2}lt;/p>

Again substituting u = 1 - y², we get

$\frac{(1 - y^2)^{\frac{1}{2}}}{\pi} + C = ∫\cos(πx) dx{2}lt;/p>

And again Substitute y = cos(πx) here,

$\frac{\left(1 - \cos(πx)^2x\right)^{\frac{1}{2}}}{π} + C = ∫\cos(πx) dx{2}lt;/p>

$\frac{\left(\sin^2(πx)\right)^{\frac{1}{2}}}{π}+C=∫\cos(πx)dx{2}lt;/p>

$\frac{\sin(πx)}{π}+C =∫\cos(πx)dx{2}lt;/p>

Hence the integral of cos pi x is sin(πx)/π. It is used in calculus to solve different integral problems.

Integral of cos(pix) 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 formula can be written as:

$∫^b_a f(x) dx = F(b) – F(a){2}lt;/p>

Let’s understand the verification of the integral of cos(πx) by using the definite integral.

Proof of integral of cos(pix) by using definite integral

To compute the antiderivative of cos pi x 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(πx) from 0 to π. For this we can write the integral as:

$∫^π_0 \cos(πx)dx =\left|\frac{\sin(πx)}{π}\right|^π_0{2}lt;/p>

Now, substituting the limit in the given function.

$∫^π_0 \cos(πx)dx=\frac{\sin(π^2)}{π}-\frac{\sin(0)}{π}{2}lt;/p>

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

$∫^π_0 \cos(πx)dx = 0{2}lt;/p>

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

$∫^{\frac{π}{2}}_0 \cos(πx)dx = \left|\frac{\sin(πx)}{π}\right|^{\frac{π}{2}}_0{2}lt;/p>

Now,

$∫^{\frac{π}{2}}_0 \cos(πx)dx=\frac{\sin\frac{π^2}{2}}{π}-\frac{\sin(0)}{π}{2}lt;/p>

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

$∫^{\frac{π}{2}}_0 \cos(πx) = 0{2}lt;/p>

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