Class 12 Maths Integration Formulas — All Types With Solved Examples | Padhaao Blog

In short: Every integration formula you need for Class 12 boards and competitive exams — organized by type. Basic integrals, special forms, substitution rules, By Parts (ILATE), Partial Fractions, definite integral properties, and 5 fully solved examples. No theory. Just formulas, methods, and worked solutions you can revise in 15 minutes.

Basic Integration Formulas

#	Function	Integral	Condition
1	∫xⁿ dx	xⁿ⁺¹/(n+1) + C	n ≠ -1
2	∫(1/x) dx	ln\|x\| + C	x ≠ 0
3	∫eˣ dx	eˣ + C	—
4	∫aˣ dx	aˣ/ln(a) + C	a > 0, a ≠ 1
5	∫sin x dx	-cos x + C	—
6	∫cos x dx	sin x + C	—
7	∫tan x dx	ln\|sec x\| + C	—
8	∫cot x dx	ln\|sin x\| + C	—
9	∫sec x dx	ln\|sec x + tan x\| + C	—
10	∫cosec x dx	ln\|cosec x - cot x\| + C	—
11	∫sec²x dx	tan x + C	—
12	∫cosec²x dx	-cot x + C	—
13	∫sec x tan x dx	sec x + C	—
14	∫cosec x cot x dx	-cosec x + C	—
15	∫1/√(1-x²) dx	sin⁻¹x + C	\|x\| < 1
16	∫-1/√(1-x²) dx	cos⁻¹x + C	\|x\| < 1
17	∫1/(1+x²) dx	tan⁻¹x + C	—
18	∫-1/(1+x²) dx	cot⁻¹x + C	—

Special Integration Formulas (Standard Forms)

#	Form	Result
1	∫1/(x² + a²) dx	(1/a) tan⁻¹(x/a) + C
2	∫1/(x² - a²) dx	(1/2a) ln\|(x-a)/(x+a)\| + C
3	∫1/(a² - x²) dx	(1/2a) ln\|(a+x)/(a-x)\| + C
4	∫1/√(x² + a²) dx	ln\|x + √(x²+a²)\| + C
5	∫1/√(x² - a²) dx	ln\|x + √(x²-a²)\| + C
6	∫1/√(a² - x²) dx	sin⁻¹(x/a) + C
7	∫√(x² + a²) dx	(x/2)√(x²+a²) + (a²/2)ln\|x+√(x²+a²)\| + C
8	∫√(x² - a²) dx	(x/2)√(x²-a²) - (a²/2)ln\|x+√(x²-a²)\| + C
9	∫√(a² - x²) dx	(x/2)√(a²-x²) + (a²/2)sin⁻¹(x/a) + C

Integration by Substitution — Key Patterns

Pattern in Integrand	Substitution	Result Pattern
f'(x)/f(x)	Let u = f(x)	ln\|f(x)\| + C
f'(x) × [f(x)]ⁿ	Let u = f(x)	[f(x)]ⁿ⁺¹/(n+1) + C
√(a² - x²)	x = a sinθ	Uses sin⁻¹
√(x² + a²)	x = a tanθ	Uses sec/log
√(x² - a²)	x = a secθ	Uses sec/log
1/(a + b cos x) or 1/(a + b sin x)	t = tan(x/2)	Converts to rational in t

Common Substitution Examples

Integral	Substitution	Answer
∫sin³x cos x dx	u = sin x, du = cos x dx	sin⁴x/4 + C
∫eˢⁱⁿˣ cos x dx	u = sin x, du = cos x dx	eˢⁱⁿˣ + C
∫tan x dx	u = cos x, du = -sin x dx	-ln\|cos x\| + C = ln\|sec x\| + C
∫2x/(1+x²) dx	u = 1+x², du = 2x dx	ln(1+x²) + C

Integration by Parts (ILATE Rule)

The Formula

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

ILATE Priority Order

Priority	Category	Examples	Choose as 'u'
1st (I)	Inverse Trigonometric	sin⁻¹x, cos⁻¹x, tan⁻¹x	First choice
2nd (L)	Logarithmic	ln x, log x	Second choice
3rd (A)	Algebraic	x, x², x³, polynomials	Third choice
4th (T)	Trigonometric	sin x, cos x, tan x	Fourth choice
5th (E)	Exponential	eˣ, aˣ	Last choice (choose as v)

Key By-Parts Results to Remember

Integral	Result
∫x eˣ dx	eˣ(x - 1) + C
∫x² eˣ dx	eˣ(x² - 2x + 2) + C
∫x sin x dx	-x cos x + sin x + C
∫x cos x dx	x sin x + cos x + C
∫ln x dx	x ln x - x + C
∫x ln x dx	(x²/2) ln x - x²/4 + C
∫eˣ[f(x) + f'(x)] dx	eˣ f(x) + C
∫eˣ(sin x) dx	eˣ(sin x - cos x)/2 + C
∫eˣ(cos x) dx	eˣ(sin x + cos x)/2 + C

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Partial Fractions

Types of Partial Fractions

Form of Denominator	Partial Fraction Form	Example
(x-a)(x-b)	A/(x-a) + B/(x-b)	1/[(x-1)(x-2)]
(x-a)²	A/(x-a) + B/(x-a)²	1/(x-1)²
(x-a)(x-b)(x-c)	A/(x-a) + B/(x-b) + C/(x-c)	1/[(x-1)(x-2)(x-3)]
(x-a)(x²+bx+c)	A/(x-a) + (Bx+C)/(x²+bx+c)	1/[(x-1)(x²+1)]
(x-a)²(x-b)	A/(x-a) + B/(x-a)² + C/(x-b)	1/[(x-1)²(x-2)]

Steps for Partial Fractions

Step	Action
1	Factor the denominator completely
2	Write the partial fraction form based on type
3	Multiply both sides by the denominator
4	Substitute convenient values of x to find A, B, C
5	Integrate each partial fraction separately

Definite Integral Properties

#	Property	Formula
1	Limits swap	∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx
2	Same limits	∫ₐᵃ f(x)dx = 0
3	Splitting	∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx
4	Variable change	∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx
5	Zero to a	∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx
6	Even function	∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx (if f(-x) = f(x))
7	Odd function	∫₋ₐᵃ f(x)dx = 0 (if f(-x) = -f(x))
8	Period 2a	∫₀²ᵃ f(x)dx = 2∫₀ᵃ f(x)dx (if f(2a-x) = f(x))
9	Period 2a	∫₀²ᵃ f(x)dx = 0 (if f(2a-x) = -f(x))
10	Leibniz Rule	d/dx[∫ₐˣ f(t)dt] = f(x)

Solved Example 1 — Basic Integral

Evaluate: ∫(3x² + 2x - 5) dx

Step	Working
Apply power rule to each term	∫3x² dx + ∫2x dx - ∫5 dx
Integrate	3(x³/3) + 2(x²/2) - 5x + C
Simplify	x³ + x² - 5x + C

Solved Example 2 — Substitution

Evaluate: ∫ 2x/(1 + x²)² dx

Step	Working
Let u = 1 + x²	du = 2x dx
Rewrite	∫ du/u² = ∫ u⁻² du
Integrate	u⁻¹/(-1) + C = -1/u + C
Substitute back	-1/(1 + x²) + C

Solved Example 3 — By Parts (ILATE)

Evaluate: ∫ x eˣ dx

Step	Working
Choose u = x (A), dv = eˣ dx (E)	du = dx, v = eˣ
Apply formula: uv - ∫v du	x·eˣ - ∫eˣ dx
Integrate	x·eˣ - eˣ + C
Factor	eˣ(x - 1) + C

Solved Example 4 — Partial Fractions

Evaluate: ∫ 1/[(x-1)(x-2)] dx

Step	Working
Write partial fractions	1/[(x-1)(x-2)] = A/(x-1) + B/(x-2)
Multiply by (x-1)(x-2)	1 = A(x-2) + B(x-1)
Put x = 1	1 = A(-1) → A = -1
Put x = 2	1 = B(1) → B = 1
Integrate	∫[-1/(x-1) + 1/(x-2)] dx
Result	-ln\|x-1\| + ln\|x-2\| + C = ln\|(x-2)/(x-1)\| + C

Solved Example 5 — Definite Integral with Property

Evaluate: ∫₀^(π/2) sin²x dx

Step	Working
Use property: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx	I = ∫₀^(π/2) sin²(π/2 - x) dx = ∫₀^(π/2) cos²x dx
So I = ∫₀^(π/2) sin²x dx = ∫₀^(π/2) cos²x dx	—
Add: 2I = ∫₀^(π/2) (sin²x + cos²x) dx	2I = ∫₀^(π/2) 1 dx = π/2
Solve	I = π/4

Method Selection Guide

If the integrand looks like...	Use this method
xⁿ, sin x, cos x, eˣ (simple)	Direct formula
f'(x)/f(x) or f'(x)[f(x)]ⁿ	Substitution
Product of two different types (x·eˣ, x·sin x, ln x)	By Parts (ILATE)
Polynomial / factored polynomial	Partial Fractions
√(a²-x²), √(x²+a²), √(x²-a²)	Trigonometric substitution
sin²x, cos²x, sin³x cos²x	Use identities first, then integrate
1/(a+b sin x) or 1/(a+b cos x)	Half-angle substitution: t = tan(x/2)
Even/Odd function with symmetric limits	Use definite integral properties

Frequently Asked Questions

Q1. How many integration formulas do I need to memorize? You need about 18 basic formulas and 9 special forms. The rest (By Parts results, Partial Fraction forms) come from applying methods. Focus on memorizing the basic table first — everything builds on it.

Q2. What is the ILATE rule and when do I use it? ILATE stands for Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. It tells you which function to choose as 'u' in integration by parts. Always pick the function that appears earlier in ILATE as 'u'.

Q3. How do I know which method to use? Look at the integrand: (1) Simple function → direct formula, (2) composite function with its derivative present → substitution, (3) product of two different types → by parts, (4) rational function → partial fractions. The Method Selection Guide table above covers all cases.

Q4. Which definite integral properties are most important? Properties 4, 5, 6, and 7 are most frequently used in board exams. Property 5 (∫₀ᵃ f(x) = ∫₀ᵃ f(a-x)) is used in nearly every definite integral problem involving trigonometric functions.

Q5. How many marks does integration carry in the board exam? Integration (Chapter 7) typically carries 10-12 marks in the CBSE Class 12 Maths paper — usually one 5-mark question, one 3-mark question, and 1-2 MCQs. It is the single highest-weightage chapter.

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Class 12 Maths Integration Formulas — All Types With Solved Examples