Schooling · Class 12 · Mathematics · Integrals · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-II
Integrals: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- An antiderivative of f(x) is any…
- Standard results for powers…
- Substitution (change of variable)…
- Integration by parts follows from…
- Partial fractions break a rational…
- The definite integral ∫ from a to b…
- Geometrically, the definite integral…
- Formulas and facts
- ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ −1
- ∫ (1/x) dx = ln |x| + C
- ∫ eˣ dx = eˣ + C
- ∫ aˣ dx = aˣ / ln a + C, a > 0, a ≠ 1
- ∫ sin x dx = −cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec²x dx = tan x + C
- Watch out for
- Forgetting the constant C in…
- Applying the power rule when n = −1
- Dropping the differential du or dx…
- Wrong choice of u in integration by…
- Sign errors when evaluating F(b) −…
What this chapter is about
Integration is the reverse process of differentiation. Where differentiation breaks a function into its instantaneous rate of change, integration pieces together those rates to recover the original function or to find accumulated quantities such as area, volume or total distance. A Class 12 student meets this chapter because many problems in physics, economics and geometry require summing infinitely many infinitesimally small parts, and the integral is the precise tool for that task.
Key ideas
- An antiderivative of f(x) is any function F(x) such that dF/dx = f(x). The indefinite integral ∫ f(x) dx represents the family of all antiderivatives, written F(x) + C, where C is an arbitrary constant.
- Standard results for powers, exponentials, trigonometric and inverse trigonometric functions form the foundation: for instance, ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C when n ≠ −1.
- Substitution (change of variable) simplifies an integral by replacing a complicated inner function with a single new variable.
- Integration by parts follows from the product rule: ∫ u dv = uv − ∫ v du, useful when the integrand is a product of two different types of function.
- Partial fractions break a rational function into simpler pieces that can each be integrated using logarithms or inverse-tangent formulas.
- The definite integral ∫ from a to b of f(x) dx equals F(b) − F(a), where F is any antiderivative of f. This is the Fundamental Theorem of Calculus.
- Geometrically, the definite integral gives the net signed area between the curve y = f(x) and the x-axis from x = a to x = b.
- Properties such as ∫ from a to b = −∫ from b to a, and ∫ from a to b + ∫ from b to c = ∫ from a to c, help evaluate and manipulate definite integrals.
Formulas and facts to remember
- Formula: ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ −1 · Meaning: Power rule for integration
- Formula: ∫ (1/x) dx = ln |x| + C · Meaning: Integral of reciprocal gives natural log
- Formula: ∫ eˣ dx = eˣ + C · Meaning: Exponential unchanged on integration
- Formula: ∫ aˣ dx = aˣ / ln a + C, a > 0, a ≠ 1 · Meaning: General exponential base a
- Formula: ∫ sin x dx = −cos x + C · Meaning: Sine integrates to negative cosine
- Formula: ∫ cos x dx = sin x + C · Meaning: Cosine integrates to sine
- Formula: ∫ sec²x dx = tan x + C · Meaning: Derivative of tan x is sec²x
- Formula: ∫ 1 / (1 + x²) dx = arctan x + C · Meaning: Leads to inverse tangent
- Formula: ∫ 1 / √(1 − x²) dx = arcsin x + C · Meaning: Leads to inverse sine
- Formula: ∫ u dv = uv − ∫ v du · Meaning: Integration by parts rule
- Formula: ∫ from a to b of f(x) dx = F(b) − F(a) · Meaning: Fundamental Theorem of Calculus
Common mistakes
- Forgetting the constant C in indefinite integrals → always add + C because any constant vanishes on differentiation.
- Applying the power rule when n = −1 → use ∫ (1/x) dx = ln |x| + C instead.
- Dropping the differential du or dx during substitution → keep track; the differential tells you how to replace dx with du.
- Wrong choice of u in integration by parts, leading to a harder integral → try LIATE priority: Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential; pick u from earlier in this list.
- Sign errors when evaluating F(b) − F(a) → substitute upper limit first, then subtract the value at the lower limit.
Quick revision
- Integration undoes differentiation; add C for the general antiderivative.
- Master the standard table: powers, 1/x, eˣ, sin x, cos x, sec²x, 1/(1 + x²), 1/√(1 − x²).
- Substitution replaces a composite function with a simpler variable.
- Integration by parts: ∫ u dv = uv − ∫ v du; use LIATE to pick u.
- Fundamental Theorem: ∫ from a to b of f(x) dx = F(b) − F(a).