Integral Of Cos(X^2)

Integral of cos(x^2) along with its formula and proof with examples. Also learn how to calculate integration of cos(x^2) with step by step examples.

Alan Walker-

Published on 2023-04-14

Introduction to integral of cos(x^2)

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area between two curves, 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.

Integral calculator 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(x^2) integral by using different integration techniques.

What is the integral of cos(x^2)?

The integral of cos(x^2) is an antiderivative of cos(x2) function which is done by using Taylor’s series expansion. It is also known as the reverse derivative of cos(x2) 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(x^2) formula

The formula of the cos(x^2) integral contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(cos x2)dx. In mathematical form, the integral of cos x2 is:

$∫\cos x^2dx=x-\frac{x^5}{5×2!}+\frac{x^9}{9×4!}+\frac{x^{13}}{13×6!}+...+ C$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. In the above formula, we can replace x^2 by x^3 to find the integration of cos(x^3).

How to find the integral cos(x)^2?

The antiderivative of cos x 2 is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of cos x^2 by using:

  1. Taylor’s series expansion
  2. Definite integral

Integral of cos x2 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 derivative of a function that is complex to solve. Since cos(x^2) is impossible to integrate by using formal integration. Therefore, we will use Taylor’s series to find the integral of cos(x^2).

Proof of integral cos(x^2) by using Taylor’s Series

Since we know that the integration is the reverse of the derivative. Therefore, we can evaluate integral of cos(x)^2 with respect to x by using Taylor’s series. For this, we have to first assume the sine series that is,

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

We can use the above series in the integral of sin x to calculate the integral of cos x2. Then,

$I = ∫\cos x^2dx$

Substituting the series of cosx, we get,

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

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

$∫\cos x^2dx=x-\frac{x^5}{5×2!}+\frac{x^9}{9×4!}+\frac{x^{13}}{13×6!}+ ...+ C$

Hence the above equation is the cos(x^2) integral by using Taylor’s series.

Integral of cos x2 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:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integral of cos x2 by using the definite integral.

Proof of integral of cos x2 by using definite integral

To compute the integral of cos x square by using a definite integral, we can use the interval from 0 to π or 0 to π/2. Let’s evaluate integral of cos(x)^2 with respect to x from 0 to π. For this we can write the integral as:

$∫^π_0 \cos x^2dx=\left|x-\frac{x^5}{5×2!}+\frac{x^9}{9×4!}+\frac{x^{13}}{13×6!}+ ...\right|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \cos x^2 dx=π-\frac{π^5}{5×2!}+\frac{π^9}{9×4!}+\frac{π^{13}}{13×6!}+...- 0$

The remaining terms are:

$∫^π_0 \cos x^2 dx = π - \frac{π^5}{20}+\frac{π^9}{216}+\frac{π^{13}}{9360}+ …$

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

$∫^{\frac{π}{2}}_0 \cos x^2dx=\left|x-\frac{x^5}{5×2!}+\frac{x^9}{9×4!}+\frac{x^{13}}{13×6!}+...\right|^{\frac{π}{2}}_0$

$∫^{\frac{π}{2}}_0 \cos x^2dx = x-\frac{\pi^5}{5×2!}+\frac{\pi^9}{9×4!}+\frac{\pi^{13}}{13×6!}+...-0$

The remaining terms are:

$∫^{\frac{π}{2}}_0 \cos x^2 dx=π-\frac{π^5}{160×2!}+\frac{π^9}{4608×4!} +\frac{π^{13}}{106496×6!}+ …$

Hence it is the calculation of the integral of cos x2 by using a definite integral.

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