Integration by Parts: Indefinite Integrals

Integration by parts is a method for evaluating integrals involving the product of two terms, a function and a differential.

The formula is derived by applying the product rule of differentiation to the product of two functions.

Integrate both sides with respect to x. By substituting standard notations for differentials and rearranging terms, the integration by parts formula is obtained.

For instance, when integrating x multiplied by a cosine function, the integrand is split into two components. Typically, the function that simplifies upon differentiation is chosen as u, and the other as dv.

The function u is differentiated, and dv is integrated. These are then substituted into the integration by parts formula to get the final result.

This method plays a role in many analyses. For instance, consider an AC circuit where the current is a product of two functions.

To calculate the voltage across the capacitor, the current must be integrated. By selecting one function to differentiate and the other to integrate, the integration by parts method can be used to evaluate the result efficiently.