How do you integrate #f(x)=sinx/x# using the quotient rule?

Question

Emily Ammerman · Accepted Answer

# int sinx/xdx = Si(x) + C # # int sinx/xdx = Si(x) + C # where #Si(x)# is a special function known as the Sine Integral This is higher degree level mathematics, so I suspect your question should have been how do you differentiate #f(x)# !!

Scarlett Allison · Answer

undefined To integrate $ f(x) = \frac{\sin(x)}{x} $ using the quotient rule, first differentiate the function $ \frac{\sin(x)}{x} $ with respect to $ x $ to get the derivative, then integrate the result.

1. Differentiate $ \frac{\sin(x)}{x} $ using the quotient rule:
$$ \frac{d}{dx}\left(\frac{\sin(x)}{x}\right) = \frac{x \cdot \cos(x) - \sin(x)}{x^2} $$

2. Now integrate the result:
$$ \int \frac{x \cdot \cos(x) - \sin(x)}{x^2} \, dx $$

This integral can be solved by separating it into two integrals:

$$ \int \frac{x \cdot \cos(x)}{x^2} \, dx - \int \frac{\sin(x)}{x^2} \, dx $$

The first integral can be simplified by canceling out $ x $ from numerator and denominator:

$$ \int \frac{\cos(x)}{x} \, dx $$

This integral can be solved using techniques such as integration by parts or specialized methods for handling integrals of the form $ \int \frac{\cos(x)}{x} \, dx $.

The second integral, $ \int \frac{\sin(x)}{x^2} \, dx $, is known as the sine integral and does not have a simple elementary antiderivative. It can be expressed in terms of special functions like the Si(x) function.

Therefore, integrating $ f(x) = \frac{\sin(x)}{x} $ using the quotient rule leads to a more complex expression, involving integrals that may not have simple closed-form solutions.