Which method is used to integrate the product of two functions?

The method used to integrate the product of two functions is Integration by Parts. This technique is derived from the product rule for differentiation and is particularly useful when dealing with integrals that involve a product of two functions that cannot be easily integrated together in a straightforward manner.

The formula for Integration by Parts is given by:

∫ u dv = uv - ∫ v du

In this formula, 'u' is one function that you choose to differentiate, and 'dv' is another function that you choose to integrate. By choosing these parts wisely, you can simplify the integral on the right-hand side, making it easier to solve.

This method is especially effective when one of the functions, after differentiation, becomes simpler or when the integral of the second function, after integration, is more manageable. A common example is integrating products like x * e^x or ln(x) * x, where straightforward integration won't yield results.

In contrast, Integration by Substitution is more suitable for simplifying integrals where a substitution can be made to transform the integral into a more recognizable form. Partial Fraction Integration is used when dealing with rational functions to break them down into simpler fractions for easier integration. Definite Integration refers to the evaluation of an integral over a specific interval and