MathematicsChapter 3: Integration
Integration By Parts
Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
Learning objectives
- State the rule for integration by parts.
- Solve a integration by parts problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
Mathematical form
dv=uv-integral v du
dx=x ex-ex+C.
Where / symbols
- 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 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
- 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.
Worked example
- Worked example:
- Integral x ex dx=x ex-ex+C.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Integration by parts
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Integration By Parts
Sticky Notes · Formula and Worked Revision
Core Idea
- Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
Rule / Formula
- Integral u dv=uv-integral v du; choose u so differentiation simplifies it.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Integral x ex dx=x ex-ex+C.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- 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.
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