MathematicsChapter 3: Integration
Substitution With Limits
For definite integrals, either change limits to u-values or return fully to x before applying old limits.
Learning objectives
- State the rule for substitution with limits.
- Solve a substitution with limits problem step by step.
- Check the result, restrictions, notation, and units.
Definition
For definite integrals, either change limits to u-values or return fully to x before applying old limits.
Where / symbols
- Substitution with limits
- For definite integrals, either change limits to u-values or return fully to x before applying old limits.
CONCEPT AND RULE
For definite integrals, either change limits to u-values or return fully to x before applying old limits.
METHOD
For substitution with limits, 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
Integral01 2x(x2+1) dx=[ufrac22]12=32.
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 substitution with limits
- 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:
- Integral01 2x(x2+1) dx=[ufrac22]12=32.
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
- Substitution with limits
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Substitution With Limits
Sticky Notes · Formula and Worked Revision
Core Idea
- For definite integrals, either change limits to u-values or return fully to x before applying old limits.
Rule / Formula
- For definite integrals, either change limits to u-values or return fully to x before applying old limits.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Integral01 2x(x2+1) dx=[ufrac22]12=32.
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
- For definite integrals, either change limits to u-values or return fully to x before applying old limits.
- Worked pattern: Integral01 2x(x2+1) dx=[ufrac22]12=32.
- Final check: domain, sign, notation, units, and reasonableness.
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