In the integral ∫ x^3 ln(x) dx, what is the choice for u when applying integration by parts?

When applying integration by parts, the choice of ( u ) typically depends on selecting the function that, when differentiated, simplifies the integral. In the case of the integral ( ∫ x^3 \ln(x) , dx ), choosing ( u = \ln(x) ) is strategic for several reasons.

First, differentiating ( \ln(x) ) yields ( du = \frac{1}{x} dx ), which is a relatively simple form to work with. This simplification is beneficial because the logarithm function decreases the complexity of the integral once we substitute it back in during the integration by parts process.

On the other hand, if we were to choose ( u = x^3 ), then ( du = 3x^2 dx ), which potentially complicates the integration process due to the high polynomial degree that could lead to handling a more complex integral. Additionally, ( x^3 ) does not simplify the integral as well as ( \ln(x) ) does when we consider the resulting ( dv ) after differentiation.

Thus, choosing ( u = \ln(x) ) is indeed the most appropriate selection for maximizing the efficiency of the integration by parts method in this scenario.