Introduction to integral of cos(x^3)

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 integrals. 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 the cos(x3) integral by using different integration techniques.

What is the integral of cos(x^3)?

The cos(x^3) integral is an antiderivative of the cos(x^3) function which is done by using Taylor’s series expansion. It is also known as the reverse derivative of the cos x^3 function, a trigonometric identity.

The sine function is the ratio of the opposite side to the hypotenuse of a triangle which is written as:

cos = adjacent side/hypotenuse

Integral of cos(x³) formula

The formula of the integral of sin contains the integral sign, coefficient of integration, and the function as cos. It is denoted by ∫(cos x³)dx. In mathematical form, the integral of cos x³ is:

$∫\cos x^3dx=\frac{x^4}{4}-\frac{x^7}{7×2!}+\frac{x^{13}}{13×4!}+\frac{x^{19}}{19×6!} + ....+ C$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Replace cos(x^3) by cos(x^2) to evaluate the integral of cos(x2). 

How to evaluate integral of cos(x)^3 with respect to x?

The integral of cosx^3 is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to integrate cosx^3 by using:

1. Taylor’s series expansion
2. Definite integral

Integration of cos x^3 by using Taylor’s Series

Taylor’s series is an infinite sum of terms that are expressed in terms of a function’s derivative. It can be used to calculate the derivative of a function that is complex to solve. Since cos(x³) is impossible to integrate by using formal integration. Therefore, we will use Taylor’s series to find the cos(x^3) integral.

Proof of integral of cos x^3 by using Taylor’s Series

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the antiderivative of cos(x^3) by using Taylor’s series. For this, we have to first assume the sine series that is,

$\cos x =1–\frac{x^2}{2!}+\frac{x^4}{4!}–\frac{x^6}{6!} + …$

We can use the above series in the cos x^3 integration to calculate the integral of cos x³. Then,

$I = ∫\cos x^3dx$

Substituting the series of cosx, we get,

$I = ∫\left[1 –\frac{(x^3)^2}{2!}+\frac{(x^3)^4}{4!}–\frac{(x^3)^6}{6!} + … \right] dx$

Now we can easily integrate these terms to get the integral cos(x^2). Therefore,

$∫\cos x^3dx =x -\frac{x^7}{7×2!}+\frac{x^{13}}{13×4!}+\frac{x^{19}}{19×6!} + ....+ C$

Hence the above equation is the cosx^3 integral by using Taylor’s series. This method can also be used to calculate the integral of cos x/x.

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

Proof of integral of cosx^3 by using definite integral

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

$∫^π_0 \cos x^3dx = \left|x-\frac{x7}{7×2!}+\frac{x^{13}}{13×4!}+\frac{x^{19}}{19×6!} + ...\right|^\pi_0$

Now, substitute the limit in the given function.

$∫^π_0 \cos x^3dx=\left[π -\frac{π^7}{7×2!}+\frac{π^{13}}{13×4!}+\frac{π^{19}}{19×6!}+...\right]-0$

The remaining terms are:

$∫^π_0 \cos x^3dx = π -\frac{π^7}{28}+\frac{π^{13}}{312}+\frac{π^{19}}{13680} + …$

Which is the calculation of the definite integral of cos x3. Now to calculate the integral of cos x3 between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$∫^{\frac{π}{2}}_0 \cos x^3 dx = \left|x-\frac{x^7}{28}+\frac{x^{13}}{312}+\frac{x^{19}}{13680} + ...\right|^{\frac{π}{2}}_0$

$∫^{\frac{π}{2}}_0 \cos x^3 dx=π - \frac{(π/2)^7}{28} +\frac{(π/2)^{13}}{312} + \frac{(π/2)^{19}}{13680} +… -0$

The remaining terms are:

$∫^{\frac{π}{2}}_0 \cos x^3 dx = π -\frac{(π/2)^7}{28}+\frac{(π/2)^{13}}{312} +\frac{(π/2)^{19}}{13680} +…$

Hence it is the calculation of the integral of cos(x^3) by using a definite integral.