When integrating by parts, which formula is typically used?

The formula used for integrating by parts is derived from the product rule of differentiation and is expressed as the integral of a product of functions. The correct expression, ∫ u dv = uv - ∫ v du, indicates that you can transform an integral of the product of two functions into a simpler form.

In this formula, the integral of the first function (u) multiplied by the differential of the second function (dv) can be expressed as the product of u and v minus the integral of v multiplied by the differential of u (du). This technique is particularly useful when you have a product of functions and integrating one of them is more straightforward after performing the integration by parts.

The other options present variations that either misapply or misunderstand the foundational principles of integration by parts. For instance, the second option incorrectly states a positive sign instead of a negative, while the remaining choices pertain to basic integral forms not directly related to the method of integration by parts. Therefore, knowing and applying the correct formula is crucial for solving integrals effectively in calculus.