Integration formulas are the standard results used to find the antiderivative of a function, written in general form as ∫f'(x) dx = f(x) + C. They cover algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions, and form the working set for solving problems in Class 11, Class 12, and competitive exams like JEE.
What Is Integration?
Integration is the inverse operation of differentiation — given a function's derivative, integration recovers the original function (called the antiderivative). An indefinite integral leaves the result in general form with a constant C; a definite integral evaluates the result between two limits a and b.
The constant C exists because differentiation removes constants. Any function f(x) + C has the same derivative as f(x), so integration produces a family of curves rather than a single result.
Integration Formulas Cheat Sheet (Quick Reference)
The table below maps integrand patterns to their standard integrals. Use it as a recognition tool: identify the form of your integrand, then apply the matching formula.
If your integrand looks like... | The integral is... | Method/notes |
|---|---|---|
xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C | Power rule |
1/x | ln|x| + C | Power rule fails at n = −1 |
eˣ | eˣ + C | Direct |
aˣ | aˣ/ln a + C | Direct |
ln x | x ln x − x + C | Use integration by parts |
sin x | −cos x + C | Direct |
cos x | sin x + C | Direct |
sec²x | tan x + C | Direct |
csc²x | −cot x + C | Direct |
sec x · tan x | sec x + C | Direct |
csc x · cot x | −csc x + C | Direct |
tan x | ln|sec x| + C | Substitution |
cot x | ln|sin x| + C | Substitution |
sec x | ln|sec x + tan x| + C | Standard result |
csc x | −ln|csc x + cot x| + C | Standard result |
1/√(1 − x²) | sin⁻¹x + C | Inverse trig |
1/(1 + x²) | tan⁻¹x + C | Inverse trig |
1/(|x|√(x² − 1)) | sec⁻¹x + C | Inverse trig |
1/(a² + x²) | (1/a) tan⁻¹(x/a) + C | Generalized form |
1/√(a² − x²) | sin⁻¹(x/a) + C | Generalized form |
1/(x² − a²) | (1/2a) ln|(x − a)/(x + a)| + C | Special integral |
1/(a² − x²) | (1/2a) ln|(a + x)/(a − x)| + C | Special integral |
1/√(x² − a²) | ln|x + √(x² − a²)| + C | Special integral |
1/√(x² + a²) | ln|x + √(x² + a²)| + C | Special integral |
f(x) · g(x) | uv − ∫v du | Integration by parts |
P(x)/Q(x) (rational) | Decompose first | Partial fractions |
Basic Integration Formulas
Power Rule and Constant Rules
Formula | Notes |
|---|---|
∫xⁿ dx = xⁿ⁺¹/(n+1) + C | Valid for all n ≠ −1 |
∫(1/x) dx = ln|x| + C | The exception to the power rule |
∫k dx = kx + C | k is any constant |
∫k · f(x) dx = k · ∫f(x) dx | Constants pass through the integral |
∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx | Sum and difference rule |
The power rule fails at n = −1 because xⁿ⁺¹/(n+1) becomes x⁰/0, which is undefined. The integral of 1/x is ln|x| + C, derived from the fact that d/dx(ln|x|) = 1/x.
Exponential and Logarithmic Formulas
Formula | Notes |
|---|---|
∫eˣ dx = eˣ + C | The function that integrates to itself |
∫aˣ dx = aˣ/ln a + C | a > 0, a ≠ 1 |
∫ln x dx = x ln x − x + C | Derived using integration by parts |
∫log_a x dx = (x log_a x) − (x/ln a) + C | Change-of-base form |
Trigonometric Integration Formulas
Standard Trigonometric Integrals
Every trigonometric integration formula is a derivative read backward. Memorize 6 derivative pairs and the corresponding 12 integration formulas follow.
Derivative | Integration formula |
|---|---|
d/dx(sin x) = cos x | ∫cos x dx = sin x + C |
d/dx(cos x) = −sin x | ∫sin x dx = −cos x + C |
d/dx(tan x) = sec²x | ∫sec²x dx = tan x + C |
d/dx(cot x) = −csc²x | ∫csc²x dx = −cot x + C |
d/dx(sec x) = sec x · tan x | ∫sec x · tan x dx = sec x + C |
d/dx(csc x) = −csc x · cot x | ∫csc x · cot x dx = −csc x + C |
Integrals of tan x, cot x, sec x, csc x
Formula |
|---|
∫tan x dx = ln|sec x| + C |
∫cot x dx = ln|sin x| + C |
∫sec x dx = ln|sec x + tan x| + C |
∫csc x dx = −ln|csc x + cot x| + C |
These four are not derivative-pair formulas. Each is derived using a specific substitution or algebraic manipulation, which is why they sit outside the standard six.
Trigonometric Formulas with Linear Argument (ax + b)
When the argument inside a trig function is ax + b instead of x, divide the result by the coefficient of x.
Formula |
|---|
∫sin(ax + b) dx = −(1/a) cos(ax + b) + C |
∫cos(ax + b) dx = (1/a) sin(ax + b) + C |
∫sec²(ax + b) dx = (1/a) tan(ax + b) + C |
∫csc²(ax + b) dx = −(1/a) cot(ax + b) + C |
∫sec(ax + b) tan(ax + b) dx = (1/a) sec(ax + b) + C |
∫csc(ax + b) cot(ax + b) dx = −(1/a) csc(ax + b) + C |
Inverse Trigonometric Integration Formulas
Basic Inverse Trig Integrals
Formula |
|---|
∫1/√(1 − x²) dx = sin⁻¹x + C |
∫1/(1 + x²) dx = tan⁻¹x + C |
∫1/(|x|√(x² − 1)) dx = sec⁻¹x + C |
These are the integration counterparts of the derivatives of sin⁻¹x, tan⁻¹x, and sec⁻¹x.
Generalized Inverse Trig Formulas (with a)
Formula |
|---|
∫1/(a² + x²) dx = (1/a) tan⁻¹(x/a) + C |
∫1/√(a² − x²) dx = sin⁻¹(x/a) + C |
∫1/(|x|√(x² − a²)) dx = (1/a) sec⁻¹(x/a) + C |
The constant a generalizes the basic forms — set a = 1 and they collapse back to the basic versions.
Special Integrals (Standard Forms)
Quadratic Denominator Forms
Formula |
|---|
∫1/(x² − a²) dx = (1/2a) ln|(x − a)/(x + a)| + C |
∫1/(a² − x²) dx = (1/2a) ln|(a + x)/(a − x)| + C |
∫1/(x² + a²) dx = (1/a) tan⁻¹(x/a) + C |
Quadratic Radical Forms
Formula |
|---|
∫1/√(x² − a²) dx = ln|x + √(x² − a²)| + C |
∫1/√(a² − x²) dx = sin⁻¹(x/a) + C |
∫1/√(x² + a²) dx = ln|x + √(x² + a²)| + C |
∫√(x² − a²) dx = (x/2)√(x² − a²) − (a²/2) ln|x + √(x² − a²)| + C |
∫√(x² + a²) dx = (x/2)√(x² + a²) + (a²/2) ln|x + √(x² + a²)| + C |
∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C |
Hyperbolic Integration Formulas
Hyperbolic functions appear in advanced calculus, engineering applications, and JEE-level problems involving catenaries, special relativity, and certain differential equations.
Formula |
|---|
∫sinh x dx = cosh x + C |
∫cosh x dx = sinh x + C |
∫tanh x dx = ln(cosh x) + C |
∫coth x dx = ln|sinh x| + C |
∫sech²x dx = tanh x + C |
∫csch²x dx = −coth x + C |
∫sech x · tanh x dx = −sech x + C |
∫csch x · coth x dx = −csch x + C |
Methods of Integration
Integration by Substitution
Substitution is used when the integrand contains a function and its derivative. It converts the integral into a simpler form by replacing a function of x with a single variable.
Formula: If u = g(x), then ∫f(g(x)) · g'(x) dx = ∫f(u) du
Example: ∫2x · cos(x²) dx
Let u = x², so du = 2x dx
The integral becomes ∫cos(u) du = sin(u) + C
Substituting back: sin(x²) + C
Standard substitutions for radical forms:
Integrand contains | Substitute |
|---|---|
√(a² − x²) | x = a sin θ |
√(a² + x²) | x = a tan θ |
√(x² − a²) | x = a sec θ |
Integration by Parts
Integration by parts handles integrals of products of two functions. It is the integral counterpart of the product rule of differentiation.
Formula: ∫u · v dx = u · ∫v dx − ∫(u' · ∫v dx) dx
Equivalently, ∫u dv = uv − ∫v du.
Choosing u — the ILATE rule. When two functions are multiplied, choose u in the order Inverse, Logarithmic, Algebraic, Trigonometric, Exponential. The function higher in ILATE becomes u; the other becomes dv.
Letter | Function type | Example |
|---|---|---|
I | Inverse trigonometric | sin⁻¹x |
L | Logarithmic | ln x |
A | Algebraic | x², x³ |
T | Trigonometric | sin x, cos x |
E | Exponential | eˣ |
Example: ∫x · eˣ dx
u = x (algebraic, higher in ILATE), dv = eˣ dx
du = dx, v = eˣ
∫x · eˣ dx = x · eˣ − ∫eˣ dx = x · eˣ − eˣ + C = eˣ(x − 1) + C
Integration by Partial Fractions
Used when the integrand is a rational function P(x)/Q(x) where the degree of P(x) is less than the degree of Q(x). Decompose the rational function into simpler fractions, then integrate each.
The 5 standard cases:
Denominator form | Decomposition |
|---|---|
(x − a)(x − b), a ≠ b | A/(x − a) + B/(x − b) |
(x − a)² | A/(x − a) + B/(x − a)² |
(x − a)(x − b)(x − c) | A/(x − a) + B/(x − b) + C/(x − c) |
(x − a)²(x − b) | A/(x − a) + B/(x − a)² + C/(x − b) |
(x − a)(x² + bx + c), non-factorable quadratic | A/(x − a) + (Bx + C)/(x² + bx + c) |
Example: ∫(3x + 2)/((x − 1)(x + 2)) dx
Decompose: (3x + 2)/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2)
Solving gives A = 5/3, B = 4/3
∫(3x + 2)/((x − 1)(x + 2)) dx = (5/3) ln|x − 1| + (4/3) ln|x + 2| + C
Definite Integration Formulas and Properties
The Fundamental Theorem of Calculus
If F(x) is the antiderivative of f(x), then:
∫ₐᵇ f(x) dx = F(b) − F(a)
Every formula in the cheat sheet works for definite integrals. The only difference: substitute the limits at the end and subtract.
Properties of Definite Integrals
Property | Statement |
|---|---|
P0 | ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt |
P1 | ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx; in particular, ∫ₐᵃ f(x) dx = 0 |
P2 | ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx |
P3 | ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx |
P4 | ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx |
P5 | ∫₀²ᵃ f(x) dx = ∫₀ᵃ f(x) dx + ∫₀ᵃ f(2a − x) dx |
P6 (even/odd) | ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f(−x) = f(x); equals 0 if f(−x) = −f(x) |
Common Mistakes
A few errors recur across student work, regardless of grade level.
Forgetting +C on indefinite integrals. Indefinite integrals always carry a constant of integration. Skipping it is the most common source of lost marks on Class 12 and JEE papers.
Treating ∫(1/x) dx as a power rule case. The power rule xⁿ⁺¹/(n+1) breaks at n = −1 because the denominator becomes zero. ∫(1/x) dx = ln|x| + C, not x⁰/0.
Sign errors on ∫sin x dx. The integral of sin x is −cos x + C, not cos x. The negative sign appears because d/dx(cos x) = −sin x, so the antiderivative carries the opposite sign.
Choosing the wrong u in integration by parts. ILATE exists for a reason. Picking u as the exponential when an algebraic factor is present (e.g., choosing u = eˣ in ∫x · eˣ dx) creates an integral more complex than the original.
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