Integration Formulas Class 12: Complete List with Methods and Examples

Integrals is the biggest chapter of Class 12 Maths — 8 to 10 marks in boards and a regular JEE topic. Integration is easy once you know the standard results and can recognise which method fits. Here is the full list.

Standard integrals (memorise)

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫(1/x) dx = ln|x| + C
• ∫eˣ dx = eˣ; ∫aˣ dx = aˣ/ln a
• ∫sin x = −cos x; ∫cos x = sin x; ∫sec²x = tan x; ∫cosec²x = −cot x
• ∫sec x tan x = sec x; ∫cosec x cot x = −cosec x
• ∫tan x = ln|sec x|; ∫cot x = ln|sin x|; ∫sec x = ln|sec x + tan x|; ∫cosec x = ln|cosec x − cot x|

Special integrals

• ∫dx/(x² + a²) = (1/a) tan⁻¹(x/a)
• ∫dx/(x² − a²) = (1/2a) ln|(x − a)/(x + a)|
• ∫dx/(a² − x²) = (1/2a) ln|(a + x)/(a − x)|
• ∫dx/√(a² − x²) = sin⁻¹(x/a)
• ∫dx/√(x² ± a²) = ln|x + √(x² ± a²)|
• ∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a)
• ∫√(x² ± a²) dx = (x/2)√(x² ± a²) ± (a²/2) ln|x + √(x² ± a²)|

Methods

1. Substitution: if the integrand has f(x) and f′(x), put t = f(x). Example: ∫2x/(1 + x²) dx, t = 1 + x² → ∫dt/t = ln(1 + x²) + C.

2. By parts (ILATE): ∫u v dx = u∫v dx − ∫(u′∫v dx) dx. Choose u in the order Inverse, Logarithmic, Algebraic, Trigonometric, Exponential. Example: ∫x eˣ dx = x eˣ − eˣ + C. Special form: ∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C.

3. Partial fractions: for rational functions with degree of numerator less than denominator. Example: ∫dx/((x−1)(x−2)) = ∫[−1/(x−1) + 1/(x−2)] dx = ln|(x−2)/(x−1)| + C.

4. Trigonometric forms: use 1 − cos2x = 2sin²x for ∫sin²x, and product-to-sum for ∫sin ax cos bx.

Definite integrals: properties that save time

• ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx (the “king” property — used in most 5-mark questions)
• ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even; 0 if f is odd
• ∫₀²ᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f(2a − x) = f(x); 0 if f(2a − x) = −f(x)

Example: I = ∫₀^(π/2) sin x/(sin x + cos x) dx. Apply king property: I = ∫₀^(π/2) cos x/(sin x + cos x) dx. Add: 2I = ∫₀^(π/2) 1 dx = π/2 → I = π/4.

Common mistakes

• Forgetting + C in indefinite integrals.
• Wrong choice of u in by parts — follow ILATE.
• Not changing limits after substitution in definite integrals.

Revise the trig identities you will need: Trigonometry identities list.

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Frequently asked questions

What is ILATE rule?

The order for choosing the first function in integration by parts: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential.

How many marks is Integrals in Class 12 boards?

Around 8–10 marks including definite integrals and applications.

What is the king property of definite integrals?

∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx. Adding the two forms often simplifies the integrand.