\( \int \frac { } { 3 x } \)

To solve the integral \( \int \frac{1}{3x} \, dx \), we can factor out the constant from the integral. This gives us: \[ \int \frac{1}{3x} \, dx = \frac{1}{3} \int \frac{1}{x} \, dx \] Now, we know that the integral of \( \frac{1}{x} \) is \( \ln |x| + C \), where \( C \) is the constant of integration. Therefore, we can rewrite our result as: \[ \frac{1}{3} \ln |x| + C \] Now, you’ve completed the integration! When dealing with logarithmic integrals, remember that the method of substitution can often simplify things, especially when the integrand involves more complicated functions. For instance, if you had \( \int \frac{1}{3g(x)} g'(x) \, dx \), you'd simply use the substitution \( u = g(x) \) to make it easier. Also, it’s common to mix up the constants of integration. A helpful reminder is to treat \( C \) as an unknown constant, but make sure to write it whenever you’re dealing with indefinite integrals!