Integral Of Cos^3 X

Integral of cos^3 x along with its formula and proof with examples. Also learn how to calculate integration of cos^3 x with step by step examples.

Alan Walker-

Published on 2023-04-14

Introduction to integral of cos^3x

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 cubic sine. You will also understand how to compute the cos cube integral by using different integration techniques.

What is the integral of cos^3x?

The integration of cos cube x is an antiderivative of the cos^3x function which is equal to sin x –(1/3)sin3 x + c. It is also known as the reverse derivative of the cos cubed function, 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

Cos^3x formula integration

The formula of the integral of sin contains the integral sign, coefficient of integration, and the function as cos^3x. It is denoted by ∫(cos3x)dx. In mathematical form, the integral of sin^3x is:

$∫\cos^3xdx = \sin x – \frac{\sin^3x}{3}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. The above integration formula can also be used to calculate the integral of cos square if we replace cos^3x by cos^2x.

How to calculate the integral of cos^3(x)?

The integral of cos3x 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. Integration by parts
  2. Substitution method
  3. Definite integral

Integral of cos x cubic by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. Let’s discuss calculating the integration of cos cube x by using the integration by parts calculator.

Proof of integration of cos^3x by using integration by parts

Since we know that the function cosine cubes x can be written as the product of two functions. Therefore, we can integrate cos^3x by parts integration formula. For this, suppose that:

$I = \cos^3x = \cos x.\cos^2 x$

Applying the integral we get,

$I = ∫(\cos x.cos^2 x)dx$

Since the method of integration by parts is:

$∫[f(x).g(x)]dx = f(x).∫g(x)dx - ∫[f’(x).∫g(x)]dx$

Now replacing f(x) and g(x) by cos x, we get,

$I = \cos^2 x.\sin x + ∫[2\cos x\sin x.\sin x]dx$

It can be written as:

$I = \cos^2x.\sin x + 2∫[\sin^2 x.\cos x]dx$

Now by using a trigonometric identity sin2x = 1 – cos2x. Therefore, substituting the value of sin2x in the above equation, we get:

$I = \cos^2x.\sin x + 2∫\cos x(1 – \cos^2x)dx$

Integrating remaining terms,

$I = \cos^2 x.\sin x +2\sin x – 2\int cos^3xdx$

Since we know that I = cos3x

$I = \cos^2 x.\sin x +2\sin x – 2I$


$3I = \cos^2x.\sin x + 2\sin x$

For more simplification, substitute

$\cos^2x = 1 – \sin^2x$

$3I = \sin x(1 – \sin^2x) + 2\sin x$


$3I = \sin x – \sin^3x + 2\sin x$

$3I = 3\sin x – \sin^3x$

Now dividing by 3 on both sides,

$I = \sin x – \frac{\sin3x}{3}+c$

Hence the integrate cos^3x by parts is equal to,

$∫\cos^3xdx = \sin x – \frac{\sin3x}{3}+c$

This method is used to find the integration of a product of the product of two functions. It can be also used to compute the integral of cos^5x as it can be expressed as a product of two functions.

Integral of cos^3(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 integration of cos cube x by using the substitution method.

Proof of integration of cos^3xby using substitution method

To integrate cos^3x by using the substitution method, suppose that:

$I = ∫\cos^3xdx$

Suppose that we can write the above integral as:

$I = ∫(\cos x.\cos^2x)dx$

By using trigonometric identities, we can write the above equation by using cos2x = 1 – sin2x, therefore,

$I = ∫\cos x.( 1 – \sin^2x)dx$


$I = ∫(\cos x – \cos x\sin^2x)dx$

Now to evaluate the first integral, we will use the following steps,

$I_1 = ∫\cos x.dx = \sin x$

Now to evaluate the second integral,

$I_2 = -∫\cos x\sin^2x dx$

Suppose that u = sin x and du = cos x dx, then

$I_2 = -∫u^2 du$

Integrating with respect to u.

$I_2 = \frac{-u^3}{3}$

Substituting the value of u we get,

$I_2 = - \frac{\sin^3x}{3}$

Now, using the value of the first and second integral in the above equation to get the final value of the integral.

$I = \sin x – \frac{\sin^3x}{3} + c$

Hence the integration of cos^3x is verified by using the substitution method.

Integral of cos^3x 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^3(x) by using the definite integral.

Proof of integration of cos^3x by using definite integral

To integrate cos^3x by using a definite integral, we can use the interval from 0 to π/2 or 0 to π. Let’s compute the integral of sin^3x from 0 to 2π.

The indefinite integral of cos^3x can be written as:

$∫^{\frac{π}{2}}_0 \cos^3x dx = \left |\sin x – \frac{\sin^3x}{3}\right|^{\frac{π}{2}}_0$

Substituting the value of limit we get,

$∫^{\frac{π}{2}}_0 \cos^3x dx =\left[\sin \frac{π}{2} – \frac{\sin^3 \frac{π}{2}}{3}\right] – \left[\sin 0 – \frac{\sin^30}{3}\right]$

$∫^{\frac{π}{2}}_0 \cos^3x dx = 1 – \frac{1}{3} =\frac{2}{3}$

Therefore, the integral of cos3x from 0 to π/2 is

$∫^{\frac{π}{2}}_0 \cos^3x dx = \frac{2}{3}$

Which is the calculation of the definite integral of cos^3x. Now to calculate the integral of cos cube between the interval 0 to π, we just have to replace π/2 by π. Therefore,

$∫^π_0 \cos^3x dx = \left|\sin x – \frac{\sin^3x}{3}\right|^π_0$

$∫^π_0 \cos^3x dx =\left[\sin π – \frac{\sin3π}{3}\right]–\left[\sin 0 – \frac{sin^30}{3}\right]$

$∫^π_0 \cos^3x dx = 0 – 0$

$∫^π_0 \cos^3x dx = 0$

Therefore, the integral of cos3x from 0 to π is 0. You can also use our definite integral calculator with steps as it provides you a step-by-step solution in no time.

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