MathematicsChapter 3: Integration
Fundamental Theorem of Calculus
Differentiation and integration are inverse processes under continuity conditions.
Learning objectives
- State the rule for fundamental theorem of calculus.
- Solve a fundamental theorem of calculus problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Differentiation and integration are inverse processes under continuity conditions.
Formula
dt=x2+1.
Where / symbols
- Fundamental theorem of calculus
- Differentiation and integration are inverse processes under continuity conditions.
CONCEPT AND RULE
Differentiation and integration are inverse processes under continuity conditions.
METHOD
For fundamental theorem of calculus, 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
d/dx integral1x(t2+1)dt=x2+1.
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 fundamental theorem of calculus
- 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:
- d/dx integral1x(t2+1)dt=x2+1.
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
- Fundamental theorem of calculus
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Fundamental Theorem of Calculus
Sticky Notes · Formula and Worked Revision
Core Idea
- Differentiation and integration are inverse processes under continuity conditions.
Rule / Formula
- Differentiation and integration are inverse processes under continuity conditions.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- d/dx integral1x(t2+1)dt=x2+1.
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
- Differentiation and integration are inverse processes under continuity conditions.
- Worked pattern: d/dx integral_1^x(t^2+1)dt=x^2+1.
- Final check: domain, sign, notation, units, and reasonableness.
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