The integration by parts formula is used to integrate the product of two functions. It states that:

∫u dv = uv – ∫v du

where u and v are functions of x, and du/dx and dv/dx are their derivatives with respect to x.

To prove this formula, we start with the product rule:

(d/dx)(uv) = u dv/dx + v du/dx

Rearranging this equation, we get:

u dv/dx = (d/dx)(uv) – v du/dx

Multiplying both sides by dx and integrating, we get:

∫u dv = ∫[(d/dx)(uv) – v du/dx] dx

Using the integral of a derivative rule, we can simplify the first term on the right-hand side:

∫u dv = uv – ∫v du/dx dx

Since du/dx is the derivative of u, we can replace it with du:

∫u dv = uv – ∫v du

This is the integration by parts formula.

To use the integration by parts formula to integrate a specific function, we choose u and dv such that the integral of v du is easier to evaluate than the integral of u dv. We then compute u, du/dx, v, and dv/dx, substitute them into the formula, and evaluate the integral.