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Integration by Parts
Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
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Integration by Parts
What is it?
Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
In simple words
- Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
- Worked pattern: Integral x e^x dx=x e^x-e^x+C.
- Final check: domain, sign, notation, units, and reasonableness.
Important terms
Integration by parts: Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
What you should learn
- State the rule for integration by parts.
- Solve a integration by parts problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
CONCEPT AND RULE
Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
METHOD
For integration by parts, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
Integral x ex dx=x ex-ex+C.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve integration by parts
How it works
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Remember
- Integration by parts
- rule
- worked example
- worked example
How the ideas connect
Worked example: Integral x e^x dx=x e^x-e^x+C.
Deeper understanding
Go beyond memorising Integration by Parts. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: Integral x e^x dx=x e^x-e^x+C.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- Integration by parts
- rule
- worked example
- Integration by parts: Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
- CONCEPT AND RULE: Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
- METHOD: For integration by parts, identify the given information and required result. Select the matching de
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
For long answers on Integration by Parts: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Integration by parts
- rule
- worked example
- Integration by parts: Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
Important formulas
- Integration by parts
- rule
- worked example
Common exam traps
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- A matching-looking formula is not valid until its conditions and domain have been checked.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- Mistake
A matching-looking formula is not valid until its conditions and domain have been checked.
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Integration by parts
rule
worked example
Integration by parts: Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
- Definition
- Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
- Critical warning
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all d
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Exam Practice
Define
Q1. Define Integration by Parts.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of Integration by Parts and say why it fits.
Mistake
Q4. State one common mistake students make about Integration by Parts.
Exam note
Q5. Write a short exam note on Integration by Parts: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.