Integrals for Class 12: The Complete CBSE Guide (2026-27)

In the CBSE Class 12 Mathematics Term-II paper (2024-25 annual scheme), integrals class 12 commands approximately 15–18 marks out of 80, distributed across 1–2 long-answer questions (5–6 marks each) and 2–3 short-answer or very short-answer questions (2–4 marks each). The chapter tests both procedural fluency—can you integrate a rational function using partial fractions?—and conceptual depth, such as applying the fundamental theorem of calculus or using symmetry properties to simplify definite integrals. Historically, Question 10 or 11 in Section C is a 5-mark integration problem involving substitution or by-parts, while Section B contains a 3-mark definite integral evaluation. Because integrals also underpin the Applications of Integrals chapter (area under curves, volumes), weak integration skills cascade into lost marks elsewhere. Competitive exams like JEE Main and Advanced allocate even higher weightage, making integrals class 12 a dual investment for board and entrance preparation.

An indefinite integral, written ∫ f(x) dx, represents the most general anti-derivative of f(x). The result is a family of functions differing by a constant C. NCERT introduces standard integrals such as ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C (n ≠ -1), ∫ sin x dx = –cos x + C, and ∫ eˣ dx = eˣ + C. Students must memorize these forms because every advanced integral reduces to combinations of standard forms through algebraic manipulation or substitution. A common pitfall is omitting the constant C; in board exams, omitting C in the final answer of an indefinite integral costs you 1 mark. The NCERT textbook Chapter 7, Section 7.2, lists eighteen standard integrals. Practice involves recognizing which standard form applies after rewriting the integrand. For example, ∫ dx/(a² + x²) = (1/a) tan⁻¹(x/a) + C is a direct formula, but ∫ dx/(4 + x²) requires identifying a = 2.

Substitution is the reverse of the chain rule. When the integrand is a composite function or contains a function and its derivative, set u = g(x), compute du = g'(x) dx, and rewrite the integral in terms of u. NCERT Section 7.3 demonstrates this with examples like ∫ 2x cos(x²) dx: substitute u = x², du = 2x dx, yielding ∫ cos u du = sin u + C = sin(x²) + C. Substitution also handles integrals of the form ∫ f'(x)/f(x) dx = ln|f(x)| + C. A strategic substitution can transform a complicated rational function into a standard form. For instance, ∫ dx/(x² + 2x + 5) completes the square: x² + 2x + 5 = (x + 1)² + 4, then substitute u = x + 1, reducing to ∫ du/(u² + 4) = (1/2) tan⁻¹(u/2) + C = (1/2) tan⁻¹((x+1)/2) + C. Board examiners test recognition: which substitution simplifies the integrand fastest?

Integration by parts applies to integrals of products: ∫ u dv = uv – ∫ v du. The ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential) guides which function to choose as u. For example, in ∫ x eˣ dx, choose u = x (algebraic) and dv = eˣ dx. Then du = dx, v = eˣ, yielding x eˣ – ∫ eˣ dx = x eˣ – eˣ + C = eˣ(x – 1) + C. NCERT Section 7.6 presents several worked examples, including ∫ x cos x dx and ∫ ln x dx (here u = ln x, dv = dx). A tricky case is ∫ eˣ sin x dx, requiring two successive applications of by-parts that loop back to the original integral, which you then solve algebraically. Board papers frequently test by-parts with a twist: ∫ x² eˣ dx needs two iterations. Common mistakes include sign errors when subtracting ∫ v du and forgetting to add C.

When the integrand is a rational function P(x)/Q(x) with degree of P < degree of Q, decompose it into partial fractions. NCERT Section 7.5 covers four cases: (i) Q(x) has distinct linear factors, (ii) repeated linear factors, (iii) irreducible quadratic factors, (iv) combination. For example, (3x + 5)/((x – 1)(x + 2)) = A/(x – 1) + B/(x + 2). Solve for A and B by equating coefficients or substituting convenient x-values (cover-up method). Once decomposed, each term integrates to a standard form, typically ln|x – a| or tan⁻¹. If degree of P ≥ degree of Q, first perform polynomial long division. A 5-mark board question often gives a cubic denominator with one repeated root: Q(x) = (x – a)²(x – b). The partial fraction setup is A/(x – a) + B/(x – a)² + C/(x – b). Students frequently err in setting up the correct number of terms or in solving the system of equations for coefficients.

A definite integral ∫ₐᵇ f(x) dx yields a number representing the net signed area between f(x) and the x-axis from x = a to x = b. The fundamental theorem of calculus (NCERT Section 7.8) states: if F(x) is an anti-derivative of f(x), then ∫ₐᵇ f(x) dx = F(b) – F(a). Note that the constant C cancels, so you never write C in definite integrals. The theorem has two parts: Part 1 asserts that d/dx [∫ₐˣ f(t) dt] = f(x), and Part 2 provides the evaluation formula. CBSE board questions test both: Part 1 appears in 2-mark 'find derivative' problems, Part 2 in 4–5 mark evaluation problems. A key insight is that definite integrals do not require you to solve the indefinite integral if symmetry or properties apply (see next section). Geometrically, if f(x) is negative on [a, b], the integral is negative; total area is |∫ₐᵇ f(x) dx|.

NCERT Section 7.9 lists seven properties that drastically simplify evaluation. Property P₁: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt (dummy variable). Property P₂: ∫ₐᵇ f(x) dx = –∫ᵇₐ f(x) dx. Property P₃: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx (additivity). Property P₄: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b – x) dx (King's rule or reflection property). Property P₅: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a – x) dx. Property P₆ (even/odd): If f is even, ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx; if f is odd, ∫₋ₐᵃ f(x) dx = 0. Property P₇ (periodic): ∫₀ⁿᵃ f(x) dx = n ∫₀ᵃ f(x) dx if f has period a. Mastering these properties can reduce a 5-mark problem to two lines. For instance, ∫₀^π x sin x dx: use King's rule I = ∫₀^π (π – x) sin(π – x) dx = ∫₀^π (π – x) sin x dx. Adding I to itself yields 2I = π ∫₀^π sin x dx, so I = π [–cos x]₀^π = π(1 – (–1)) = 2π; hence I = π².

Effective integrals class 12 notes consolidate standard forms, substitution patterns, by-parts templates, and property statements onto 2–3 pages. Students should create a two-column table: left column lists the integral form (e.g., ∫ dx/(x² + a²)), right column gives the result (1/a tan⁻¹(x/a) + C) and any conditions (a > 0). Include worked examples for each technique. Annotate common errors: forgetting absolute value in ln, sign flip in by-parts, incorrect partial-fraction setup. Many toppers color-code their notes—blue for indefinite, green for definite, red for properties. Digital note-takers can use apps that render LaTeX, making formula review faster. CBSETUTOR.ai allows students to snap a photo of any integral problem from their worksheet or previous-year paper and receive an instant step-by-step solution aligned with NCERT methods, effectively creating personalized integrals class 12 notes on demand. The AI tutor costs ₹999/month for unlimited questions across all subjects and classes 6–12, with a 3-day free trial (no card required).

Analyzing past five years of CBSE board papers reveals recurring question types. (1) Evaluate ∫ (px + q)/√(ax² + bx + c) dx (5 marks, 2019, 2022): complete the square in the denominator, split numerator into derivative of denominator plus constant, then integrate. (2) Prove that ∫₀^(π/2) f(sin x) dx = ∫₀^(π/2) f(cos x) dx using properties (3 marks, 2020, 2023). (3) Integrate rational functions with cubic denominators via partial fractions (5 marks, 2018, 2021). (4) Use integration by parts to find ∫ x² cos x dx or ∫ (ln x)² dx (4 marks, 2019). (5) Evaluate ∫₋ₐᵃ xⁿ f(x²) dx by checking even/odd symmetry (2 marks, 2020, 2022). Practicing these question types from NCERT Miscellaneous Exercise and Exemplar ensures 80–90% coverage of board exam patterns. CBSETUTOR.ai's question bank includes every NCERT example, exercise problem, and exemplar question with video-style text walkthroughs, accessible 24×7 for ₹999/month.

Indefinite integrals yield a family of functions (include +C), whereas definite integrals yield a number (no C). Use indefinite integrals when the problem asks for the anti-derivative or general solution of a differential equation. Use definite integrals when computing area, distance, or any quantity over a specified interval. In practice, you often find the indefinite integral first, then apply limits if the problem is definite. However, properties of definite integrals can bypass finding the anti-derivative altogether. For example, evaluating ∫₀^(π) sin⁵x cos⁴x dx by first computing the indefinite integral is tedious; instead, recognize sin⁵x cos⁴x is an odd function in sin x and apply symmetry. Board exams test both: 3-mark questions may ask 'integrate...' (indefinite), while 4–5 mark questions say 'evaluate ∫ₐᵇ...' (definite). Always check the question wording—forgetting C in an indefinite integral answer loses 1 mark, writing C in a definite integral is technically incorrect but usually not penalized if the final numeric answer is right.

Error 1: Omitting +C in indefinite integrals (costs 1 mark). Fix: write C immediately after integrating, before simplifying. Error 2: Incorrect substitution back to original variable—students forget to replace u everywhere, leaving mixed variables. Fix: after integrating in u, substitute u = g(x) in every term. Error 3: Sign mistakes in integration by parts, especially the – sign in ∫ u dv = uv – ∫ v du. Fix: write the formula at the top of your working, tick off terms as you substitute. Error 4: Partial fractions setup errors, such as using A/(x² + 1) instead of (Ax + B)/(x² + 1) for irreducible quadratics. Fix: memorize the four cases and practice 10 examples of each. Error 5: Forgetting absolute value in ∫ 1/x dx = ln|x| + C; writing ln x loses marks if domain includes negative x. Fix: always write ln|...| unless context guarantees positivity. Error 6: Applying definite integral properties incorrectly, e.g., claiming ∫₋₁¹ x² dx = 0 because 'it looks symmetric' (x² is even, not odd; integral = 2∫₀¹ x² dx = 2/3). Fix: test f(–x) = ±f(x) algebraically before applying even/odd rules.

Below is the exhaustive formula sheet every Class 12 student must memorize. Standard integrals: ∫ 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; ∫ sin x dx = –cos x + C; ∫ cos x dx = sin x + C; ∫ sec²x dx = tan x + C; ∫ cosec²x dx = –cot x + C; ∫ sec x tan x dx = sec x + C; ∫ cosec x cot x dx = –cosec x + C; ∫ tan x dx = ln|sec x| + C; ∫ cot x dx = ln|sin x| + C; ∫ sec x dx = ln|sec x + tan x| + C; ∫ cosec x dx = ln|cosec x – cot x| + C. Inverse trigonometric forms: ∫ dx/√(a²–x²) = sin⁻¹(x/a) + C; ∫ dx/√(x²+a²) = sinh⁻¹(x/a) + C or ln|x + √(x²+a²)| + C; ∫ dx/√(x²–a²) = cosh⁻¹(x/a) + C or ln|x + √(x²–a²)| + C; ∫ dx/(a²+x²) = (1/a) tan⁻¹(x/a) + C; ∫ dx/(x²–a²) = (1/(2a)) ln|(x–a)/(x+a)| + C. Keep this list on a flashcard and review daily.

CBSETUTOR.ai is India's first 24×7 AI tutor built exclusively on the CBSE NCERT curriculum for Classes 6–12. Every integral formula, theorem proof, and NCERT example from Chapter 7 (Integrals) is embedded in the AI's knowledge base. Students can photograph any problem—whether from RD Sharma, RS Aggarwal, previous-year boards, or their school worksheet—and receive an instant, step-by-step solution that mirrors NCERT's methodology. The AI explains which technique to apply (substitution, by-parts, partial fractions, properties) and highlights common errors specific to that problem type. Parents appreciate the flat ₹999/month pricing that covers all subjects and classes 6–12, eliminating the need for multiple expensive tutors. A 3-day free trial (no credit card required) lets families test the platform on integrals class 12 problems before committing. Unlike passive video lectures, CBSETUTOR.ai is interactive: students can ask follow-up questions ('Why did you choose u = x here?') and get context-aware answers instantly. For integrals, where practice volume determines mastery, this on-demand support accelerates learning and builds exam confidence.

Step 1: Read the question twice. Identify if it is indefinite (find the anti-derivative) or definite (evaluate between limits). Step 2: Scan for keywords—'evaluate', 'prove using properties', 'hence find'—which signal the technique. Step 3: Write the relevant formula or property statement first (e.g., 'Using ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b–x) dx'); this shows the examiner your method and earns partial marks even if calculation errors occur. Step 4: Show all algebraic steps—examiners deduct marks for unjustified leaps. For partial fractions, write the decomposition setup and show solving for coefficients. Step 5: Box or underline the final answer. For definite integrals, write the numeric answer clearly; for indefinite, ensure +C is present. Step 6: Verify if time permits: differentiate your indefinite integral or plug limits into a calculator for definite integrals. Step 7: Presentation matters—use standard mathematical notation, avoid overwriting, leave margins for examiner comments. Practicing 20–25 previous years' questions under timed conditions (5 marks in 12 minutes, 3 marks in 7 minutes) builds speed and accuracy.