Integration of \(\frac{\sin(x)}{x^2}\) Explained

please integrate (sin(x))/x^2 using any elementary and special functions required

To integrate the function (\frac{\sin(x)}{x^2}), we can use integration by parts. Let's denote the integral as:

[ I = \int \frac{\sin(x)}{x^2} , dx ]

We can use integration by parts, where we let:

• ( u = \frac{\sin(x)}{x} ) (which we will differentiate)
• ( dv = \frac{1}{x} , dx ) (which we will integrate)

Now, we need to compute (du) and (v):

1. Differentiate (u): [ du = \left( \frac{\cos(x)}{x} - \frac{\sin(x)}{x^2} \right) dx ]

2. Integrate (dv): [ v = \ln|x| ]

Now, applying integration by parts:

[ I = uv - \int v , du ]

Substituting (u), (du), and (v):

[ I = \frac{\sin(x)}{x} \ln|x| - \int \ln|x| \left( \frac{\cos(x)}{x} - \frac{\sin(x)}{x^2} \right) dx ]

This integral can be quite complex, and it does not lead to a simple expression in terms of elementary functions.

However, the integral (\int \frac{\sin(x)}{x} , dx) is known as the sine integral, denoted as (\text{Si}(x)). The integral (\int \frac{\sin(x)}{x^2} , dx) does not have a simple closed form in terms of elementary functions, but it can be expressed in terms of special functions.

Thus, the integral can be expressed as:

[ \int \frac{\sin(x)}{x^2} , dx = -\frac{\cos(x)}{x} + \int \frac{\cos(x)}{x} , dx ]

The integral (\int \frac{\cos(x)}{x} , dx) is known as the cosine integral, denoted as (\text{Ci}(x)).

In conclusion, the integral (\int \frac{\sin(x)}{x^2} , dx) does not have a simple expression in terms of elementary functions, but it can be expressed in terms of the cosine integral function:

[ \int \frac{\sin(x)}{x^2} , dx = -\frac{\cos(x)}{x} + \text{Ci}(x) + C ]

where (C) is the constant of integration.