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
Integration by parts is a fundamental tool in calculus and is widely used in various fields, including