What is the integral of the function 3/x with respect to x?

The correct integral of the function ( \frac{3}{x} ) with respect to ( x ) is indeed given by ( 3\ln|x| + C ). This result arises from the fundamental rule of integration for functions of the form ( \frac{1}{x} ), which has the integral ( \ln|x| + C ).

To derive the integral, consider the integral ( \int \frac{3}{x} , dx ). Since 3 is a constant, we can factor it out of the integral:

[

\int \frac{3}{x} , dx = 3 \int \frac{1}{x} , dx

]

The integral of ( \frac{1}{x} ) is ( \ln|x| + C ). Therefore, substituting this back into our equation gives:

[

= 3(\ln|x| + C) = 3\ln|x| + 3C

]

Since ( 3C ) is still a constant (which we can denote as ( C )), we simplify to:

[

3\ln|x| + C

]