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

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

• A. 3ln|x| + C
• B. ln|3x| + C
• C. 3/x + C
• D. ln|x| + C

Answer: (A)

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: \[ 3\ln|x| + C \]

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

]