CBSE Class 12 Integration — Complete Guide with All Methods & Solved Examples

In short: Integration (Chapter 7 — Integrals) is one of the most important and highest-scoring chapters in CBSE Class 12 Maths, carrying 8-10 marks out of 80. It covers indefinite integrals (standard forms, substitution, partial fractions, by parts) and definite integrals (properties, area under curve). You can expect 2-3 questions — at least one 5-marker on integration by parts or partial fractions. This chapter requires consistent practice but rewards generously once you get the patterns.

Marks Distribution

For all Maths formulas in one place, see the CBSE Class 12 Maths formulas list. For marks weightage across all chapters, check the Maths chapter-wise marks weightage.

Standard Integrals — Must Memorize

These are the building blocks. Every integration problem ultimately reduces to one of these forms.

Special Standard Forms

Method 1: Integration by Substitution

When to use: When you spot a function and its derivative in the integrand.

Strategy: Let t = inner function, then dt = (derivative) dx.

Common substitutions:

• ∫ f(ax+b) dx → Let t = ax+b
• ∫ f(g(x)) · g'(x) dx → Let t = g(x)
• ∫ sin²x dx or ∫ cos²x dx → Use half-angle formulas
• For √(a²-x²), put x = a sin θ
• For √(x²+a²), put x = a tan θ
• For √(x²-a²), put x = a sec θ

Method 2: Integration by Partial Fractions

When to use: For rational functions P(x)/Q(x) where degree of P < degree of Q and Q(x) can be factored.

Steps:

1. Factor the denominator completely
2. Write the partial fraction form
3. Multiply both sides by the denominator
4. Find constants by substituting convenient values of x (or comparing coefficients)
5. Integrate each fraction separately

Method 3: Integration by Parts (ILATE Rule)

Formula: ∫ u·v dx = u ∫v dx - ∫[du/dx · ∫v dx] dx

ILATE Rule — choose the first function (u) in this priority order:

The function earlier in ILATE becomes u (first function), the later one becomes v (second function).

Special Formula for eˣ

∫ eˣ[f(x) + f'(x)] dx = eˣ f(x) + C

This is a board favourite. Whenever you see eˣ multiplied by a sum of a function and its derivative, the answer is instant.

Method 4: Definite Integrals — Properties

Definition: ∫ₐᵇ f(x) dx = F(b) - F(a) where F(x) is the antiderivative.

Key Properties

Board exam tip: Property P4 is the most tested. When direct integration is hard, replace x by (a-x) and add both forms to simplify.

Solved Examples

Example 1: Substitution

Q: Evaluate ∫ sin³x cos x dx

Solution: Let t = sin x → dt = cos x dx ∫ sin³x cos x dx = ∫ t³ dt = t⁴/4 + C = sin⁴x/4 + C

Example 2: Partial Fractions

Q: Evaluate ∫ (2x+1)/[(x+1)(x+2)] dx

Solution: (2x+1)/[(x+1)(x+2)] = A/(x+1) + B/(x+2)

2x+1 = A(x+2) + B(x+1)

Put x = -1: 2(-1)+1 = A(1) → A = -1 Put x = -2: 2(-2)+1 = B(-1) → B = 3 (since -3 = -B)... wait: x = -2: -4+1 = B(-2+1) → -3 = B(-1) → B = 3... but let me verify: -3 = -B → B = 3

Hmm, let me recheck. Put x = -2: 2(-2)+1 = A(0) + B(-2+1) → -3 = -B → B = 3

∫ [-1/(x+1) + 3/(x+2)] dx = -ln|x+1| + 3ln|x+2| + C

Verification: Differentiate the answer to confirm you get the original integrand.

Example 3: Integration by Parts

Q: Evaluate ∫ x sin x dx

Solution: Using ILATE: u = x (Algebraic), v = sin x (Trigonometric)

∫ x sin x dx = x(-cos x) - ∫ 1·(-cos x) dx = -x cos x + ∫ cos x dx = -x cos x + sin x + C

Example 4: Special eˣ Formula

Q: Evaluate ∫ eˣ(sin x + cos x) dx

Solution: Here f(x) = sin x, f'(x) = cos x. Using ∫ eˣ[f(x) + f'(x)] dx = eˣ f(x) + C:

∫ eˣ(sin x + cos x) dx = eˣ sin x + C

Example 5: Definite Integral — Property P4

Q: Evaluate ∫₀^(π/2) log(sin x) dx... This is a classic, but let's do a simpler board-level one.

Q: Evaluate ∫₀^(π/2) sin²x dx

Solution: Using the property ∫₀^(π/2) sinⁿx dx = ∫₀^(π/2) cosⁿx dx:

∫₀^(π/2) sin²x dx = ∫₀^(π/2) (1-cos 2x)/2 dx = (1/2)[x - sin 2x/2]₀^(π/2) = (1/2)[(π/2 - 0) - (0 - 0)] = π/4

Example 6: Partial Fractions — Repeated Factor

Q: Evaluate ∫ x/[(x-1)²(x+2)] dx

Solution: x/[(x-1)²(x+2)] = A/(x-1) + B/(x-1)² + C/(x+2)

x = A(x-1)(x+2) + B(x+2) + C(x-1)²

Put x = 1: 1 = B(3) → B = 1/3 Put x = -2: -2 = C(9) → C = -2/9 Put x = 0: 0 = A(-1)(2) + (1/3)(2) + (-2/9)(1) 0 = -2A + 2/3 - 2/9 = -2A + 4/9 A = 2/9

∫ [(2/9)/(x-1) + (1/3)/(x-1)² + (-2/9)/(x+2)] dx = (2/9)ln|x-1| - (1/3)(1/(x-1)) - (2/9)ln|x+2| + C

Board Exam Important Questions

1. Evaluate ∫ (3x+5)/(x²+3x-18) dx using partial fractions. (4 marks)
2. Evaluate ∫ x² sin⁻¹x dx using integration by parts. (4 marks)
3. Evaluate ∫ eˣ[(1+sin x)/(1+cos x)] dx. (4 marks)
4. Evaluate ∫₀^π x sin x/(1+cos²x) dx using properties. (4 marks) — Classic 5-mark property question
5. Evaluate ∫ dx/(5-4cos x) using half-angle substitution. (4 marks)
6. Evaluate ∫ (x²+1)/(x²-5x+6) dx. (4 marks)
7. Evaluate ∫ √(tanx)/[sin x cos x] dx. (4 marks)
8. Prove that ∫₀^a f(x) dx = ∫₀^a f(a-x) dx and use it to evaluate ∫₀^(π/2) sin²x/(sinx + cosx) dx. (5 marks)

Frequently Asked Questions

How many marks does Integration carry?

8-10 marks out of 80 in the theory paper. Including the Application of Integrals (Chapter 8) which adds another 4-5 marks, calculus as a whole carries about 15 marks.

What is the ILATE rule?

ILATE stands for Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. When doing integration by parts, choose the first function (u) according to this priority — the one that appears first in ILATE becomes u.

How many standard integrals should I memorize?

All the ones listed in the NCERT table — roughly 20-25 standard forms. The special forms (involving √(x²±a²)) are especially important for 3-5 mark problems.

What is the most common 5-mark question?

Either a partial fractions problem with a quadratic denominator, or a definite integral using property P4 (∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx). Practice at least 10 of each type.

How do I decide which method to use?

Look at the integrand: if you see a fraction with factorable denominator → partial fractions. If you see a product of two different types (like x·sinx) → by parts. If you see a function and its derivative → substitution. If none are obvious, try substitution first.

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