integrationbyparts

Integration by parts is a technique used in calculus to find the integral of a product of functions. It is derived from the product rule for differentiation. The product rule states that the derivative of a product of two functions, u and v, is given by d/dx(uv) = u(dv/dx) + v(du/dx). Rearranging this equation, we get uv'(x) = d/dx(u(x)v(x)) - u'(x)v(x). Integrating both sides with respect to x gives the integration by parts formula: integral of uv' dx = uv - integral of u'v dx.

The formula is often written in a more compact form: integral of u dv = uv - integral of v du. Here, u and dv are chosen such that the integral of v du is simpler to evaluate than the original integral. The key to successfully applying integration by parts lies in selecting appropriate functions for u and dv. Common strategies for choosing u include LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) or ILATE, where the function that appears first in the mnemonic is typically chosen as u.

Integration by parts is a fundamental tool in calculus and is widely used in various fields, including physics, engineering, and statistics, for solving a wide range of integration problems that cannot be solved using simpler methods like substitution. It can also be applied multiple times if the resulting integral still requires further simplification.