Which of the following represents an integral that requires integration by parts?

The integral that involves the function ( x \ln(x) ) is indeed best approached using integration by parts. This method is particularly useful when integrating the product of two functions where one function can be easily differentiated and the other can be easily integrated.

In this case, we can set ( u = \ln(x) ), which would allow us to differentiate it easily, resulting in ( du = \frac{1}{x}dx ). For the other part of the product, we can set ( dv = x , dx ), which integrates to ( v = \frac{x^2}{2} ).

Now, applying the integration by parts formula:

[

\int u , dv = uv - \int v , du

]

would help us evaluate ( \int x \ln(x) , dx ) efficiently.

The other integrals presented do not require integration by parts because they involve either a simple polynomial function (like ( x^2 ) or ( x^4 )) or the exponential function ( e^x ), both of which can be integrated straightforwardly with basic integration rules. Thus, the integral ( \int x \ln(x) , dx )