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 the 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 integration 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:

cos = adjacent side/hypotenuse

Integral of cosax formula

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

The integral 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

Integration of cosax 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 cosax 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 integration 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 cos ax integration, that is:

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

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

$∫\cos(ax)dx = \frac{\sin(ax)}{a} + c{2}nbsp;

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

Cos(ax) integral 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 cos ax by using substitution method

To prove the integration of cosax by using the substitution method, suppose that:

$y = \cos(ax)$

Differentiating with respect to x,


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.


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.


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

Then the above left-hand side integral becomes,

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


Since the power rule of integration is

$∫x^ndx =\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,


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

Again substituting u = 1 - y², we get

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

And again Substitute y = cos(ax) here,

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



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.

Cos ax integration by using the 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 cos(ax) by using the definite integral.

Proof of integration of cos ax by using definite integral

To compute the integral 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 integration of cosax 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 cosax. 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,

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


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

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.


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.

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