Question - Indefinite Integration of a Constant Divided by a Variable

The integral given in the image is:

This integral represents the antiderivative of the function \( \frac{3}{x} \) with respect to \( x \). When integrating functions like this, we can use the power rule of integration. However, in this case, since 3 is a constant and does not depend on \( x \), we can factor it out of the integral. This results in the following equivalent expression:

Now we have to integrate \( \frac{1}{x} \), which is a standard integral. The antiderivative of \( \frac{1}{x} \) is known to be the natural logarithm of \( x \), denoted as \( \ln{|x|} \), where the absolute value signs are used because the natural logarithm is only defined for positive numbers.

With the constant 3 factored out, the integration proceeds as:

Here, \( C \) represents the constant of integration. The constant arises because when taking an indefinite integral, we are looking for all possible antiderivatives of the function, and the family of antiderivatives differs by a constant.

Therefore, the solution to the integral as given in the image is:

This is the general form of the antiderivative for the given integrand.