# Integral Of Cos (Ax)

Integral of cos(ax) along with its formula and proof with examples. Also learn how to calculate integration of cos(ax) with step by step examples.

Alan Walker-

Published on 2023-04-14

## Introduction to integral of cos(ax)

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(ax) integral by using different integration techniques.

## What is the integral of cos(ax)?

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

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

### Integral of cos(ax) formula

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

$∫\cos(ax)dx = \frac{\sin(ax)}{a} + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. For any value of an in the above formula, we can calculate the integral of cos(ax). For example, if a=2, the above formula will give the integral of cos(2x).

## How to calculate the integral of cos(ax)?

The integral of cos(ax) 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(ax) 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(ax) by using derivatives.

### Proof of integral of cos(ax) by using derivatives

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

In derivative, we know that,

$\frac{d}{dx}[sin ax] = a\cos(ax)$

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

$\frac{d}{dx}[\sin(ax)] = a\cos(ax)$

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

$∫\cos(ax)dx = \frac{\sin(ax)}{a} + c$

Also, learn to calculate the integral of cos(4x) by using derivatives.

## Integral of cos(ax) 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 sin by using the substitution method.

### Proof of Integral of cos(ax) by using substitution method

To prove the integral of cos(ax) by using the substitution method, suppose that:

$y = \cos(ax)$

Differentiating with respect to x,

$\frac{dy}{dx}=-a\sin(ax)$

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

$dy = -a\sin(ax).dx$

By trigonometric identities, we know that sin(ax) = √1 - cos²ax. Then the above equation becomes,

$dy = -a\sqrt{1 - cos²ax}.dx$

Now, substituting the value of cos2ax, such as:

$dy = -a\sqrt{1 – y^2}.dx$

Multiplying both sides by cos(ax),

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

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

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

Integrating on both sides by applying integral,

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

Now by using the u-substitution calculator, suppose that 1 - y² = u.

Then

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

Then the above left-hand side integral becomes,

$-\frac{1}{2}∫\frac{-1}{a\sqrt u}du =∫\cos(ax)dx$

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

Since the power rule of integration is

$∫x^ndx =\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,

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

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

Again substituting u = 1 - y², we get

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

And again Substitute y = cos(ax) here,

$\frac{\left(1 - \cos^2(ax)\right)^{1/2}}{a} + C =∫\cos(ax)dx$

$\frac{\left(\sin^2(ax)\right)^{1/2}}{a}+C=∫\cos(ax)dx$

$\frac{\sin(ax)}{a}+C=∫\cos(ax)dx$

Hence the integral of cos(ax) is sin(ax)/a. We can find the integral of cos(5x) by replacing a with 5 in the cos(ax) integral.

## Integral of cos(ax) 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(ax) by using the definite integral.

### Proof of integral of cos(ax) by using definite integral

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

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

Now, substitute the limit in the given function.

$∫^π_0 \cos(ax)dx=\frac{sin(aπ)}{a}-\frac{sin(0)}{a}$

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

$∫^π_0 \cos(ax)dx =0$

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

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

Now,

$∫^{\frac{π}{2}}_0 \cos(ax)dx=\frac{\sin \frac{aπ}{2}}{a}-\frac{\sin(0)}{a}$

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

$∫^{\frac{π}{2}}_0 \cos(ax)dx= \frac{1}{a}-0=\frac{1}{a}$

Therefore, the definite integral of cos(ax) is equal to 1/a. You can also use our definite integration calculator to avoid long-term tricky calculations.

## FAQ’s

### Can you integrate sin2x?

Integration of sin2x means finding the integral of the function sin2x. Integral of sin2x can be written as ∫ sin2x dx. Here, we need to find the indefinite integral of sin2x. So, the integration of sin2x results in a new function with arbitrary constant C.

### What is integral 2x?

The integration of 2x in calculus is equal to x square plus the constant of integration which is symbolically written as ∫2x dx = x2 + C, where ∫ is the symbol of the integral, dx shows that the integration of 2x is with respect to the variable x and C is the constant of integration.