--- title: Integration by Parts - Explained With Formula & Examples description: Integration by parts reverses the product rule. Choose u and dv, then apply the formula step by step. url: https://astra-ai.co/videos/integrals/integration-by-parts/ locale: en type: video-chapter keywords: - integration by parts - indefinite integral - product rule reverse - uv formula published: 2026-10-01T09:47:04.572Z updated: 2026-10-04T21:59:22.268Z alternates: lt: https://astra-ai.co/lt/videos/integralai/integravimas-dalimis/ fr: https://astra-ai.co/fr/videos/integrales/integration-par-parties/ es: https://astra-ai.co/es/videos/integrales/integracion-por-partes/ pl: https://astra-ai.co/pl/videos/calki/calkowanie-przez-czesci/ my: https://astra-ai.co/my/videos/my-integrals/my-18b31615/ sl: https://astra-ai.co/sl/videos/integrali/integracija-po-delih/ it: https://astra-ai.co/it/videos/integrali/integrazione-per-parti/ fi: https://astra-ai.co/fi/videos/integraalit/osittaisintegrointi/ da: https://astra-ai.co/da/videos/integraler/partiel-integration/ hr: https://astra-ai.co/hr/videos/integrali/integracija-po-dijelovima/ pt: https://astra-ai.co/pt/videos/integrais/integracao-por-partes/ nl: https://astra-ai.co/nl/videos/integralen/integratie-per-onderdeel/ el: https://astra-ai.co/el/videos/olokliromata/oloklirosi-kata-meri/ bg: https://astra-ai.co/bg/videos/integrali/integratsiya-po-chasti/ cs: https://astra-ai.co/cs/videos/integraly/integrace-po-castech/ id: https://astra-ai.co/id/videos/integral/integrasi-parsial/ hu: https://astra-ai.co/hu/videos/integralok/parcialis-integralas/ et: https://astra-ai.co/et/videos/integraalid/ositi-integreerimine/ vi: https://astra-ai.co/vi/videos/vi-topic-02/vi-chapter-45/ uk: https://astra-ai.co/uk/videos/uk-topic-02/uk-chapter-45/ no: https://astra-ai.co/no/videos/integraler/delvis-integrasjon/ sv: https://astra-ai.co/sv/videos/integraler/partiell-integration/ ro: https://astra-ai.co/ro/videos/integrale/integrare-prin-parti/ sk: https://astra-ai.co/sk/videos/integraly/kapitola-039/ lv: https://astra-ai.co/lv/videos/integrali/integresana-pa-dalam/ sr: https://astra-ai.co/sr/videos/integrali/poglavlje-039/ tr: https://astra-ai.co/tr/videos/integraller/bolum-039/ zh-TW: https://astra-ai.co/zh-TW/videos/integrals/chapter-18b31615/ th: https://astra-ai.co/th/videos/integrals/bot-45-inthikret-doi-suan/ lo: https://astra-ai.co/lo/videos/lo-integrals/lo-18b31615/ ms: https://astra-ai.co/ms/videos/integrals/bab-45-matematik-asas/ km: https://astra-ai.co/km/videos/integrals/chapter-18b31615/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-002/shuxue-zhang-045/ es-AR: https://astra-ai.co/es-AR/videos/integrales/capitulo-39/ br: https://astra-ai.co/br/videos/integrais/capitulo-matematica-045/ es-CL: https://astra-ai.co/es-CL/videos/integrales/capitulo-39/ en-IN: https://astra-ai.co/en-IN/videos/integrals/chapter-18b31615/ mx: https://astra-ai.co/mx/videos/integrales/capitulo-39/ ms-BN: https://astra-ai.co/ms-BN/videos/kamiran/bab-18b31615/ es-PE: https://astra-ai.co/es-PE/videos/integrales/capitulo-video-18b31615/ en-PH: https://astra-ai.co/en-PH/videos/integrals/chapter-18b31615/ pt-TL: https://astra-ai.co/pt-TL/videos/integrais/capitulo-de-video-039/ ja: https://astra-ai.co/ja/videos/integrals-ja/39/ ko: https://astra-ai.co/ko/videos/integrals-ko/39/ en: https://astra-ai.co/videos/integrals/integration-by-parts/ de: https://astra-ai.co/de/videos/integralrechnung/partielle-integration-per-partes/ --- # Integration by Parts > Integration by parts reverses the product rule for integrals of products. This chapter shows how to choose u and dv and apply the formula. ## INTRODUCTION TO A SPECIAL METHOD OF INTEGRATION Integration by parts (Latin: per partes) is one of the more important techniques for calculating indefinite integrals. It is based on the product rule for differentiation and allows for the solving of integrals where the integrand is a product of two expressions that are not independently easy to integrate. ## FORMULA FOR INTEGRATION BY PARTS Let f(x)g(x) be the product of two differentiable functions. Then: ∫f(x)g′(x) dx = f(x)g(x) − ∫f′(x)g(x) dx. In a more commonly used form, the formula is typically written as: **∫u dv = u \* v − ∫v du**, where: - **u** is the function that we differentiate (so du = u′ dx), - **dv** is the function (or part of the integrand including dx) that we integrate (so v = ∫dv). The success of the method relies on a smart choice of 'u' and 'dv' such that the integral ∫v du becomes simpler than the original integral ∫u dv. ## CHOOSING FUNCTIONS U AND DV For choosing the functions, the **LIATE** rule is often considered as a guideline (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential functions), where functions higher on the list are preferred choices for 'u'. This is because the derivatives of logarithmic and inverse trigonometric functions are algebraic, and algebraic functions become simpler when differentiated repeatedly. Exponential and trigonometric functions often cycle when differentiated or integrated. ## EXAMPLE: ∫x \* e^x dx Choose: - u = x → du = dx - dv = e^x dx → v = ∫e^x dx = e^x Apply the formula: ∫x \* e^x dx = x \* e^x − ∫e^x \* dx \= x \* e^x − e^x + C Solution: ∫x \* e^x dx = e^x(x − 1) + C. ## EXAMPLE: ∫ln(x) dx Here we use a special form where ln(x) is treated as 'u', and 'dv' is taken as 'dx'. We write ∫ln(x) dx as: ∫ln(x) \* 1 dx. Choose: - u = ln(x) → du = (1/x) dx - dv = 1 dx → v = ∫1 dx = x It follows: ∫ln(x) dx = x \* ln(x) − ∫x \* (1/x) dx \= x \* ln(x) − ∫1 dx \= x \* ln(x) − x + C. ## CONCLUSION Integration by parts is a key method for solving integrals of products of functions. With the rule ∫u dv = u\*v − ∫v du, we transform a more difficult integral into an easier one, where the success of the method depends on the correct choice of functions 'u' and 'dv'. This technique is often used multiple times in succession or in combination with other integration methods.