Integrals
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Indefinite Integrals and Standard Forms
Part of: Indefinite Integration
Integration undoes differentiation, returning a whole family of answers shifted by an arbitrary constant. Power, trig, exponential and log rules reverse their derivative twins, and sums split into separate integrals.
Why it matters. Every area, volume and growth problem starts by finding the right antiderivative.
Formulas
- Integral of x to the n is x to the (n+1) over (n+1), plus C
- Sums split; constants factor out
- Always append plus C for indefinite answers
Key words
- Antiderivative
- : A function whose derivative is the given integrand.
- Constant of integration
- : The arbitrary C collecting all shifted answers.
- Integrand
- : The function sitting inside the integral sign.
- Indefinite integral
- : The whole family of antiderivatives of a function.
Try it in two minutes
- Differentiate a guess and compare with the integrand.
- Match the integrand to a standard reversed rule.
- Append plus C to collect the whole family.
Try at home. Integrate 6x² − 4x + 9 and check by differentiating back.
Substitution and Special Forms
Part of: Indefinite Integration
Substitution renames an inner function so the chain rule reverses cleanly. Quadratic denominators and root forms match fixed templates with inverse trig or log answers, fractions split into partial pieces, and products split by parts with the ILATE order picking u.
Why it matters. Nearly every board integral yields to one of these four tools.
Formulas
- Substitute t for the inner function and dt for its derivative
- Denominator x² + a² gives (1/a)tan⁻¹(x/a)
- Partial fractions need factored denominators first
Key words
- Substitution
- : Renaming an inner function to reverse the chain rule.
- Partial fractions
- : Splitting one fraction into simpler additive pieces.
- Integration by parts
- : Splitting a product into u and v pieces with a fixed rule.
- ILATE order
- : The preference list picking u: inverse, log, algebraic, trig, exponential.
Try it in two minutes
- Spot the inner function and rename it t.
- Match denominators against the template list.
- Pick u by ILATE before splitting products.
Try at home. Integrate x·cos(x²) by substitution and check by differentiating.
Definite Integrals and FTC
Part of: Definite Integrals
A definite integral is a number: the antiderivative evaluated at the top limit minus the bottom one, for continuous integrands. The area function differentiates back to the integrand, and substitution rewrites the limits along with the variable.
Why it matters. Areas, totals and averages all cash out through definite values.
Formulas
- Definite value is F(top) minus F(bottom)
- Area function differentiates back to integrand
- Substitution rewrites limits too
Key words
- Definite integral
- : A single number from integrating between two limits.
- Fundamental Theorem
- : Top minus bottom of any antiderivative gives the value.
- Area function
- : The integral from a fixed start to a moving x.
- Limits of integration
- : The bottom and top fence values.
Try it in two minutes
- Find any antiderivative first.
- Subtract bottom value from top value.
- Rewrite limits when substituting.
Try at home. Evaluate ∫₀² (3x² + 1)dx and check each step.
Definite Integral Properties
Part of: Definite Integrals
Mirroring x into a − x leaves 0-to-a integrals unchanged, unlocking symmetric pairs. Odd functions cancel to zero on symmetric fences while even ones double the half. Neighbouring intervals join by additivity, and any interval splits at interior points.
Why it matters. These shortcuts evaluate in seconds what brute force cannot finish.
Formulas
- Mirror 0-to-a integrals with a minus x
- Odd cancels to zero; even doubles the half
- Join neighbours; split at interior points
Key words
- King property
- : Mirroring 0-to-a integrals with a minus x.
- Odd function
- : A function with f(−x) = −f(x), cancelling symmetric integrals.
- Even function
- : A function with f(−x) = f(x), doubling half integrals.
- Additivity
- : Joining neighbour intervals into one integral.
Try it in two minutes
- Rewrite 0-to-a integrals with the mirror a − x.
- Test the integrand for even or odd symmetry.
- Join or split intervals at shared fences.
Try at home. Evaluate ∫₋₃³ x⁵dx by symmetry alone.
What your child will practise
- Definite Integral Properties: Desert Puzzle (Clue Puzzle)
- Definite Integral Properties: Jungle Quick Fire (Quick Fire)
- Definite Integrals and FTC: Space Puzzle (Clue Puzzle)
- Indefinite Integrals and Standard Forms: Desert Quest (Flip and Match)
- Substitution and Special Forms: Space Quick Fire (Quick Fire)
- Quiz: Definite Integral Properties (8 questions)
- Quiz: Definite Integrals and FTC (8 questions)
- Quiz: Indefinite Integrals and Standard Forms (8 questions)
- Quiz: Substitution and Special Forms (8 questions)