MathematicsChapter 3: Integration
Repeated Integration By Parts
Polynomial times exponential or trigonometric factors may require the rule more than once.
Learning objectives
- State the rule for repeated integration by parts.
- Solve a repeated integration by parts problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Polynomial times exponential or trigonometric factors may require the rule more than once.
Formula
dx=ex(x2-2x+2)+C.
Where / symbols
- Repeated integration by parts
- Polynomial times exponential or trigonometric factors may require the rule more than once.
CONCEPT AND RULE
Polynomial times exponential or trigonometric factors may require the rule more than once.
METHOD
For repeated 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 x2 ex dx=ex(x2-2x+2)+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 repeated 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 x2 ex dx=ex(x2-2x+2)+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
- Repeated 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.Repeated Integration By Parts
Sticky Notes · Formula and Worked Revision
Core Idea
- Polynomial times exponential or trigonometric factors may require the rule more than once.
Rule / Formula
- Polynomial times exponential or trigonometric factors may require the rule more than once.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Integral x2 ex dx=ex(x2-2x+2)+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
- Polynomial times exponential or trigonometric factors may require the rule more than once.
- Worked pattern: Integral x^2 e^x dx=e^x(x^2-2x+2)+C.
- Final check: domain, sign, notation, units, and reasonableness.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.