Ch 07 - Integrals12th Mathematics
Definition of the Indefinite Integral
How is the indefinite integral of a function mathematically defined in relation to its derivative, and what notation denotes it?
How is the indefinite integral of a function mathematically defined in relation to its derivative, and what notation denotes it?
Integration is mathematically defined as the inverse process of differentiation.
- Definition: If , then is defined as an anti-derivative of .
- Mathematical Notation:
- The symbol represents 'the family of anti-derivatives with respect to '.
Boundary Conditions / Domain:
The function must be continuous on the interval of integration for an anti-derivative to be guaranteed by the Fundamental Theorem of Calculus.
Prism Memory Hook
A premium mnemonic for this topic is available in the Prism App.
Necessity of the Constant of Integration
Why must an arbitrary constant strictly be included when evaluating an indefinite integral?
Why must an arbitrary constant strictly be included when evaluating an indefinite integral?
- Concept: Differentiation mathematically eliminates (kills) constants because .
- Conclusion: Therefore, undoing differentiation via integration can only recover the original function up to a free additive constant.
You must strictly include in every indefinite integral answer to represent the entire family of parallel curves.
Prism Memory Hook
A premium mnemonic for this topic is available in the Prism App.
Linearity Property of Indefinite Integrals
Linearity Property of Indefinite Integrals
State the linearity property formula for splitting integrals with constants.
Component Anatomy:
- : Integrable functions on a given domain.
- : Any real constant numbers.
Boundary Conditions / Domain:
- Constants can be pulled strictly outside the integral.
- Integrals split over addition and subtraction, exactly mirroring the linearity of derivatives.
Prism Memory Hook
Constants out, Sums split.
Also in this chapter
57 more cards to study in Prism
- 4Power Rule of Indefinite Integration
- 5Standard Integral of $1/x$
- 6Standard Integrals of Exponential Functions
- 7Standard Integrals of Sine and Cosine
- 8Standard Integrals of Tangent and Cotangent
- 9Standard Integrals of Secant and Cosecant
- 10Standard Integrals of Squared Trigonometric Functions
- 11Standard Integrals of Secant-Tangent and Cosecant-Cotangent Products
- 12Standard Inverse Trigonometric Integrals
- 13Post-Integration Verification Check
- 14Log-Derivative Substitution Pattern
- 15Power-Derivative Substitution Pattern
- 16Integration by Half-Angle Identity Reduction
- 17Algorithm for Trigonometric Product-to-Sum Integrals
- 18Substitution Heuristics for $\sin^m x \cos^n x$
- 19Substitution SOP: $\tan^m x \sec^n x$ (Even Secant Power)
- 20Special Integral: $\frac{1}{x^2 + a^2}$
- 21Special Integral: $\frac{1}{x^2 - a^2}$
- 22Special Integral: $\frac{1}{a^2 - x^2}$
- 23Special Integral: $\frac{1}{\sqrt{x^2 + a^2}}$
- 24Special Integral: $\frac{1}{\sqrt{x^2 - a^2}}$
- 25Special Integral: $\frac{1}{\sqrt{a^2 - x^2}}$
- 26SOP: Completing the Square for Quadratic Denominators
- 27Numerator Splitting Algebraic Setup
- 28Gateway Check: Improper Rational Integrands
- 29Partial Fractions: Cover-Up Rule SOP
- 30Partial Fraction Template: Repeated Linear Factors
- 31Partial Fraction Template: Irreducible Quadratic Factor
- 32The Even-Power Algebraic Trap
- 33Integration by Parts Formula
- 34ILATE Priority Rule
- 35Exponential Derivative-Sum Shortcut
- 36Generalized Exponential Shortcut
- 37Definite Integral as Limit of a Sum (First Principle)
- 38The Definite Integral Bounds Trap
- 39Definite Integral Properties: P1 (Dummy Variable) & P2 (Reversal)
- 40Property P3: Splitting at an Interior Point
- 41Property P4: The Reflection Property (King Rule)
- 42Property P4 Variant: Reflection on $[0, a]$
- 43Property P5: Symmetric Interval $[-a, a]$
- 44Property P6: Doubling Rule on $[0, 2a]$
- 45Property: Integration of Periodic Functions
- 46Standard Evaluation: Integral of Log Sine/Cosine
- 47Absolute Value Trap in Trig Radicals
- 48Fixed Geometric Area: Roots of a Quadratic
- 49SOP: Eliminating Conflicting Fractional Powers
- 50SOP: Mixed Exponential Rational Integrands
- 51SOP: Sum of Squared Sines/Cosines in Denominator
- 52SOP: Linear Trigonometric Denominators
- 53Numerator Splitting: Linear Trigonometric Fractions
- 54Trigonometric Substitution for Quartic Radicals
- 55The Log-Exponential Simplification Trap
- 56SOP: Rationalisation Inside Radicals
- 57SOP Setup: The Constant Difference Trig Trick
- 58SOP Setup: Factor-Out Trick for High Power Rationals
- 59Substitution SOP: Integrals of Type $\frac{1}{L \sqrt{Q}}$
- 60Substitution SOP: Integrals of Type $\frac{1}{Q_1 \sqrt{Q_2}}$
Premium Chapter
12th Mathematics
The remaining 57 cards of Ch 07 - Integrals are available in the Aspira Prism App with full interactive study tools, spaced repetition, and memory hooks.
Get Started with Prism