Concept

# Integration by parts

Summary
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation. The integration by parts formula states: \begin{align} \int_a^b u(x) v'(x) , dx & = \Big[u(x) v(x)\Big]_a^b - \int_a^b u'(x) v(x) , dx\ & = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) , dx. \end{align} Or, letting u = u(x) and du = u'(x) ,dx while v = v(x) and dv = v'(x) , dx, the formula can be written more compactly: \int u , dv \ =\ uv - \int v , du. Mathematician Brook Taylo
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