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

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (3)

Related people

No results

Loading

Loading

Loading

Related units

No results

Related concepts (42)

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo

Series (mathematics)

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus

Related courses (39)

PHYS-210: Computational physics II

Aborder, formuler et résoudre des problèmes de physique en utilisant des méthodes numériques moyennement complexes. Comprendre
les avantages et les limites de ces méthodes (stabilité, convergence). Illustrer différents sujets de physique traités dans
d'autres cours.

ME-373: Finite element modelling and simulation

L'objectif de ce cours est d'apprendre à réaliser de manière rigoureuse et critique des analyses par éléments finis de problèmes concrets en mécanique des solides à l'aide d'un logiciel CAE moderne.

FIN-416: Interest rate and credit risk models

This course gives an introduction to the modeling of interest rates and credit risk. Such models are used for the valuation of interest rate securities with and without credit risk, the management and hedging of bond portfolios and the valuation and usage of interest rate and credit derivatives.

Related lectures (67)

Heinrich Hofmann, Swati Krishnan

Effective multimodal cancer management requires the optimal integration of diagnostic and therapeutic modalities. Radiation therapy, chemotherapy and immunotherapy, alone or in combination, are integral parts of various cancer treatment protocols. Hyperthermia at 39-45 degrees C is a potent radiosensitiser and has been shown to improve therapeutic outcomes in various tumours through its synergy with chemotherapy. Gene silencing approaches, using small interfering RNAs and microRNAs, are also being explored in clinical trials in oncology. The rapid developments in multifunctional nanoparticles provide ample opportunities to integrate both diagnostic and therapeutic modalities into a single effective cancer "theranostic" vector. Nanoparticles could extravasate passively into the tumour tissues in preference to the adjacent normal tissues by capitalizing on the enhanced permeability and retention effect. Tumour targeting might be further augmented by conjugating tumour-specific peptides and antibodies onto the surface of these nanoparticles or by activation through electromagnetic radiations, laser or ultrasound. Magnetic nanoparticles can induce hyperthermia in the presence of an alternating magnetic field, thereby multifunctionally with tumour-specific payloads empowering tumour specific radiotheranostics (for both imaging and radiotherapy), chemotherapy drug delivery, immunotherapy and gene silencing therapy. Such a (nano)bullet could realise the "magic bullet" conceived by Paul Ehrlich more than a century ago. This article discusses the various aspects of this "magic (nano)bullet" and the challenges that need to be addressed to usher in this new paradigm in modern cancer diagnostics and therapeutics. (C) 2016 Elsevier Ltd. All rights reserved.

Technological advances of the last decades enabled the development of a set of new medical techniques that differ from traditional open surgery. These techniques, referred to as minimally invasive surgery (MIS), involve complex instruments and imaging devices to reach and treat anatomical regions through small incisions or natural entry points resulting in reduced trauma and shorter recovery time. For the surgeons, MIS introduces unnatural hand-eye coordination and anatomical representation that require specific training. One alternative is the use of virtual reality (VR) simulations of MIS procedures coupled to input devices mimicking instruments used by surgeons. To achieve high-fidelity with such a computer-assisted training system, the input device also provides the sensations experienced during a procedure with force reflective technology, often referred to as haptics. This work explores the mechatronic aspects implied in the realization of such a haptic device for MIS training applications. An analysis of different MIS procedures enables to draw two categories: lumen-guided and cavity procedures. A simple classification of the required mechanism can be extracted from these observations. To fulfil the requirements, an implementation should follow a series of guidelines in terms of actuation and power transmission that are presented. The interactions between a human operator and a haptic interface involve both force and motion. Impedance control generates a force command based on the position of the haptic interface. This process, called haptic rendering, is an integrate part of the VR environment. This work proposes to extend impedance control with model-based compensation of the dynamics and non-linearities (such as friction) of the haptic device. Due to sensitivity to model uncertainties, the proposed control architecture is completed by a parallel implicit force controller. Since the computational load of VR simulation for complex anatomies is high, update rates reach 20 to 30 Hz. These frequencies are not sufficient to control a haptic device and meet the sensitivity requirements of the human sensory system. Therefore, in such system a multirate approach is introduced. A model is proposed for the dynamics of a haptic interface and its user. Based on this model, the effect of the introduced implicit force controller is studied and different interfacing techniques between the VR simulation and the haptic controller are evaluated. Since force sensing is a requirement for the implementation of the proposed implicit forcecontroller, a technique using infrared reflective sensors is proposed. A modelling method for such sensors is introduced and implementation issues for force sensing are discussed. Finally, the topics discussed in this work are applied to the development of a computer-assisted training system for Interventional Radiology.

This paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE combine together to provide a numerical tool of unprecedented quality. It is also shown that both algorithms have a much wider range of applications. A generalization of the WA algorithm, able to cope with integrands including products of Bessel and similar oscillatory functions, is described. Similarly, the original DE algorithm is adapted with exceptional results to the evaluation of the multidimensional singular integrals arising in the discretization of Integral-Equation based CEM formulations. The new possibilities of WA and DE algorithms are demonstrated through several practical numerical examples.