MathematicsChapter 3: Integration
Integration By Substitution
Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
Learning objectives
- State the rule for integration by substitution.
- Solve a integration by substitution problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
Formula
dx=(x2+1)frac44+C.
Where / symbols
- Integration by substitution
- Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
CONCEPT AND RULE
Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
METHOD
For integration by substitution, 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 2x(x2+1)3 dx=(x2+1)frac44+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 substitution
- 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 2x(x2+1)3 dx=(x2+1)frac44+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 substitution
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Integration By Substitution
Sticky Notes · Formula and Worked Revision
Core Idea
- Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
Rule / Formula
- Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Integral 2x(x2+1)3 dx=(x2+1)frac44+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
- Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
- Worked pattern: Integral 2x(x2+1)3 dx=(x2+1)frac44+C.
- Final check: domain, sign, notation, units, and reasonableness.
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