Which of the following integrals can be solved using integration by parts?

The integral of ( \int x^2 e^x , dx ) is indeed suitable for solving using integration by parts because it involves a product of a polynomial ( x^2 ) and an exponential function ( e^x ). The integration by parts technique is particularly effective here since one component, the polynomial, can be differentiated to simplify the integral, while the other component, the exponential function, remains manageable upon integration.

In integration by parts, the formula used is:

[

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

]

In this scenario, we would choose ( u = x^2 ) (which after differentiation becomes ( du = 2x , dx )) and ( dv = e^x , dx ) (which integrates to ( v = e^x )). This allows us to set up the integral in a way that makes it easier to compute.

The other integrals provided have different properties:

• The integral ( \int \cos(x) , dx ) can be solved directly using the fundamental rules of integration, leading to ( \sin(x) + C ).

• The integral ( \int \frac{1}{