Integration

• Integration as reverse differentiation and as area under a curve — definite vs indefinite integrals.
• Standard integrals from NCERT Table and linearity property ∫(af + bg) = a∫f + b∫g.
• Techniques: substitution, integration by parts ∫u dv = uv − ∫v du, and partial fractions.
• The Fundamental Theorem of Calculus linking differentiation and integration.
• Definite integrals, properties of symmetry, and evaluation of areas between curves.

Level 1 — Foundations

Verbal: Integration accumulates infinitely many infinitesimal pieces — area under y = f(x), total distance from velocity, etc.

Indefinite integral: ∫f(x) dx = F(x) + C where F′(x) = f(x). The + C captures all antiderivatives.

Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a) (FTC Part 2) when f is continuous on [a,b].

Basic integrals:

Level 2 — JEE / NEET depth

Substitution: If ∫f(g(x))g′(x) dx, put u = g(x). Classic: ∫2x cos(x²) dx → u = x².

By parts: Choose u using LIATE (Log, Inverse trig, Algebraic, Trig, Exponential). Example: ∫x eˣ dx → u = x, dv = eˣ dx.

Partial fractions (rational functions): Decompose P(x)/Q(x) when deg P < deg Q into simpler fractions before integrating.

Definite integral properties:

• ∫ₐᵇ f = −∫ᵇₐ f
• ∫ₐᵇ f + ∫ᵇᶜ f = ∫ₐᶜ f
• Even f on [−a,a]: ∫ = 2∫₀ᵃ f(x) dx
• Odd f on [−a,a]: integral = 0

Area: Between curves y = f(x), y = g(x): A = ∫|f − g| dx over intersection interval.

JEE tip: For ∫₀^(π/2) sinⁿ x cosᵐ x dx, use reduction or symmetry when n,m are integers.

Evaluate ∫ x e^(x²) dx by substitution

Step 1 — Notice derivative of x² is 2x; integrand nearly has that form.
Step 2 — Let u = x² → du = 2x dx → x dx = du/2.
Step 3 — ∫ x e^(x²) dx = ∫ e^u (du/2) = (1/2)∫ e^u du = (1/2)e^u + C.
Step 4 — Back-substitute: (1/2)e^(x²) + C.

Find area under y = x² from x = 0 to x = 2

Step 1 — Area A = ∫₀² x² dx.
Step 2 — Antiderivative: x³/3.
Step 3 — A = [x³/3]₀² = 8/3 − 0 = 8/3 square units.
Step 4 — Sanity check: rectangle 2×4 = 8; parabolic region is one-third → 8/3 ✓.

• Evaluate ∫ (3x² + 2/x) dx.
• Compute ∫₀^(π/2) sin x dx.
• Find ∫ x cos x dx using parts.
• Evaluate ∫₋₁¹ x³ dx using odd-function property.
• Stretch: Set up (do not fully solve) partial fractions for ∫ (1)/(x²−1) dx.

NCERT Chapter 7 link: Integration spans indefinite and definite integrals with properties in Section 7.2. Fundamental Theorem connects antiderivative evaluation to area — state theorem when switching from indefinite to definite.

Exam connections: Partial fractions integration is board favourite — factor denominator completely first. JEE definite integrals exploit even/odd properties on symmetric intervals to save time: ∫₋ₐᵃ odd = 0. Area between curves requires identifying upper and lower function on intersection interval.

Study strategy: Maintain LIATE priority for integration by parts. After substitution, either change limits to u-bounds or revert to x before applying FTC. Check answers by differentiating antiderivative — catches sign errors quickly.

When studying Integration within Calculus, start by listing every formula and definition on one page without looking at the textbook. Compare your list to NCERT — missing items indicate gaps to fix immediately. Work through at least two NCERT Examples for this section with steps written in full; examiners award method marks even when arithmetic slips.

For board exams (CBSE), long answers benefit from a clear structure: definition → explanation → diagram or formula → example → brief conclusion. Underline key terms. For JEE Main and NEET, prioritise conceptual traps and quick calculation paths; timed mixed quizzes of 10 questions after revision simulate exam pressure.

Cross-topic link: Coordinate geometry and vectors often combine with matrices; calculus links to physics kinematics problems.

Spaced revision: Review this note at 1 day, 3 days, and 7 days after first study. Attempt the Quick check questions closed-book, then open the Practice tab for graded reinforcement. Maintain an error log — repeated mistake patterns reveal whether the issue is concept, formula recall, or careless reading.

Diagram and terminology drill: For Mathematics, redraw key figures from memory and define every labelled part in one sentence. Vocabulary precision prevents mark loss in descriptive answers — use NCERT terms exactly as printed in the textbook.

Revision tip: Link this topic to adjacent Class 12 chapters before attempting mixed practice.

Open the Practice tab for graded questions on Integration.