Question - Comparing Integrals with Different Powers of Variable in the Denominator

\( \int \frac{3}{x} dx \neq 3 \int \frac{1}{x} dx \)

Let's solve the integral on the left side:

\( \int \frac{3}{x} dx = 3 \int \frac{1}{x} dx \)

Since \( \int \frac{1}{x} dx = \ln|x| + C \), where C is the constant of integration, we have:

\( 3 \int \frac{1}{x} dx = 3(\ln|x| + C) = 3\ln|x| + C' \)

So, the equality given in the question is incorrect, because:

\( \int \frac{3}{x} dx = 3\ln|x| + C' \)

And not:

\( 3 \int \frac{1}{x} dx = 9\ln|x| + C' \)

There seems to be a misunderstanding or typo in the provided equality.