Introduction to integral of sin(x^2)

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 calculator. 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 sin x square integration by using different integration techniques.

What is the integration of sin x square?

The sin(x^2) integral is an antiderivative of sine function which is done by using Taylor’s series expansion. It is also known as the reverse derivative of sine function which is a trigonometric identity.

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

Sin = opposite side / hypotenuse

Integration of sinx square formula

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

$∫\sin x^2dx =\frac{x^3}{3}+\frac{x^7}{7×3!}-\frac{x^{11}}{11×5!}+...+ C$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. Similarly, we can find the integral of sin x^3 by replacing x by x^3 in the Taylor's series expansion. 

How to find the sin x2 integration?

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

1. Taylor’s series expansion
2. Definite integral

Integral of sin 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 sin(x^2) is impossible to integrate by using formal integration. Therefore, we will use Taylor’s series to find the integral of sin(x^2).

Proof of sin x2 integration by using Taylor’s Series

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of sin x by using Taylor’s series. For this, we have to first assume the sine series that is,

$\sin x = x – \frac{x^3}{3!}+\frac{x^5}{5!}–\frac{x^7}{7!}+… $

We can use the above series in the integral of sin x^2 to calculate the integral of sin x^2. Then,

$I=∫\sin x^2dx$

Substituting the series of sinx, we get,

$I=∫\left[x^2–\frac{(x^2)^3}{3!}+\frac{(x^2)^5}{5!}–\frac{(x^2)^7}{7!}+ …\right]dx$

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

$∫\sin x^2dx=\frac{x^3}{3}+\frac{x^7}{7×3!}-\frac{x^{11}}{11×5!}+...+ C$

Hence the above equation is the integration of sin x² by using Taylor’s series. 

Sin x square integration by using definite integral

The definite integral calculator 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 sin x square by using the definite integral.

Proof of integration of sinx square by using definite integral

To compute the integral of sin x 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 sin x from 0 to π. For this we can write the integral as:

$∫^π_0 \sin x^2dx=\left|\frac{x^3}{3}+\frac{x^7}{7×3!}-\frac{x^{11}}{11×5!} +...\right|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \sin x^2dx=\frac{π^3}{3!}+\frac{π^7}{7×3!}-\frac{π^{11}}{11×5!}+...–0$

The remaining terms are:

$∫^π_0 \sin x^2dx=\frac{π^3}{3} +\frac{π^{7}}{7×3!}-\frac{π^{11}}{11×5!}+...$

Which is the calculation of the definite integral of sin x2. Now to calculate the sin x2 integration between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$∫^{\frac{π}{2}}_0 \sin x^2dx=\left|\frac{x^3}{3}+\frac{x^7}{7×3!}-\frac{x^{11}}{11×5!}+...\right|^{\frac{π}{2}}_0$

$∫^{\frac{π}{2}}_0 \sin x^2dx=\frac{π^3}{24}+\frac{π^7}{896×3!}-\frac{π^{11}}{22528×5!}+... –0$

The remaining terms are:

$∫^{\frac{π}{2}}_0 \sin x^2dx=\frac{π^3}{24}+\frac{π^{7}}{896×3!}-\frac{π^{11}}{22528×5!}+...$

Hence it is the calculation of integration of sin x square by using definite integral. Also calculate the integral of sin cube x by using our definite integral calculator.