When should I use integration by parts?

Use it primarily when the integrand is a product of two functions where one function becomes simpler upon differentiation (like a polynomial) and the other is easily integrable (like e^x, sin(x), cos(x)).

How do I choose 'u' and 'dv'?

A common mnemonic is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential). The function that comes earlier in this list is usually a good choice for 'u'.

Is integration by parts the same as the product rule?

It is derived from the product rule for differentiation. Essentially, integration by parts is the reverse or integral form of the product rule.

Can integration by parts be used more than once on the same problem?

Yes, this is called repeated integration by parts. It's common when integrating products involving polynomials of degree higher than one.

integration by parts

UK/ˌɪn.tɪˈɡreɪ.ʃən baɪ ˈpɑːts/US/ˌɪn.t̬əˈɡreɪ.ʃən baɪ ˈpɑːrts/

Technical (Mathematics)

My Flashcards

Definition

Meaning

A specific theorem and technique in calculus for integrating the product of two functions.

A standard method for evaluating integrals that are products, analogous to the product rule for differentiation. It can also metaphorically describe a method of problem-solving by breaking a complex product into simpler, more manageable parts.

Linguistics

Semantic Notes

The term refers strictly to a mathematical procedure. Its core concept is expressed by the formula ∫ u dv = uv - ∫ v du. Any metaphorical or extended use is highly specialized and context-dependent.

Dialectal Variation

British vs American Usage

Differences

No significant lexical or spelling differences. Pronunciation of 'parts' may vary slightly with the rhotic /r/ in American English.

Connotations

Identical technical connotations in both academic and professional mathematics contexts.

Frequency

Equally frequent in UK and US tertiary-level mathematics education and literature.

Vocabulary

Collocations

strong

use integration by partsapply integration by partsperform integration by partsformula for integration by partstechnique of integration by parts

medium

solve using integration by partsrequires integration by partsevaluate via integration by partsderive using integration by parts

weak

repeated integration by partstabular integration by partsintegration by parts ruleintegration by parts method

Grammar

Valency Patterns

to integrate [FUNCTION] by partsto apply integration by parts to [PROBLEM/INTEGRAL]Integration by parts yields [RESULT].

Vocabulary

Synonyms

Neutral

the integration by parts formulathe parts method

Weak

IBP (abbreviation)

Usage

Context Usage

Business

Not used.

Academic

Used exclusively in mathematics, physics, and engineering contexts when teaching or applying integral calculus.

Everyday

Not used.

Technical

The primary context. Used in textbooks, research papers, and problem-solving to simplify complex integrals.

Examples

By Part of Speech

verb

British English

We need to integrate x·cos(x) by parts.
You should have integrated that expression by parts.

American English

We have to integrate x·e^x by parts.
Just integrate by parts and you'll get the solution.

adjective

British English

The integration-by-parts technique is essential.
He used an integration-by-parts approach.

American English

The integration-by-parts method is on the exam.
That's a standard integration-by-parts problem.

Examples

By CEFR Level

The professor showed us how to use integration by parts for the first time.
This integral looks like it needs integration by parts.

After applying integration by parts twice, the solution became apparent.
The choice of 'u' and 'dv' is critical for the efficiency of integration by parts.

Learning

Memory Aids

Mnemonic

Remember the formula: 'Ultraviolet Voodoo' (u v minus integral of v du). Or recall it as the reverse of the product rule: differentiate u, integrate dv.

Conceptual Metaphor

A TRADE-OFF or EXCHANGE: You trade a difficult integral (∫ u dv) for a potentially simpler one (∫ v du), paying an upfront price (u*v).

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

Do not translate word-for-word as 'интеграция по частям' in general contexts; this is only correct for the mathematical term. In social/political contexts, 'integration' is 'интеграция', but 'by parts' would be nonsensical.
Avoid confusing with 'partial integration', which can be a mistranslation; the correct Russian term is 'интегрирование по частям'.

Common Mistakes

Incorrectly choosing 'u' and 'dv', leading to a more complex integral. Forgetting the minus sign in the formula. Applying it to a simple integral where a basic rule would suffice.
Mispronouncing 'integration' with a hard 'g' (like in 'get'). The 'g' is soft /dʒ/.

Practice

Quiz

Fill in the gap

To solve ∫ x sin(x) dx, you must use the technique.

Multiple Choice

What is the correct result after one application of integration by parts to ∫ x e^x dx, with u = x and dv = e^x dx?

FAQ

Frequently Asked Questions

: Use it primarily when the integrand is a product of two functions where one function becomes simpler upon differentiation (like a polynomial) and the other is easily integrable (like e^x, sin(x), cos(x)).
: A common mnemonic is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential). The function that comes earlier in this list is usually a good choice for 'u'.
: It is derived from the product rule for differentiation. Essentially, integration by parts is the reverse or integral form of the product rule.
: Yes, this is called repeated integration by parts. It's common when integrating products involving polynomials of degree higher than one.