Integrals Class 12 Maths Notes: Formulas, Properties & Examples

Key Terminology in Integrals

Integration

The process of finding a function, F(x), whose derivative is a given function, f(x). It is the reverse process of differentiation, also called anti-differentiation.

Integrand

The function that is to be integrated. In the expression ∫f(x)dx, f(x) is the integrand.

Indefinite Integral

The family of all anti-derivatives of a function f(x), denoted by ∫f(x)dx = F(x) + C, where F'(x) = f(x).

Constant of Integration (C)

An arbitrary constant added to the result of an indefinite integral. It represents the fact that the derivative of a constant is zero, so there are infinitely many possible anti-derivatives.

Definite Integral

An integral with upper and lower limits, denoted by ∫ₐᵇ f(x)dx. It represents the signed area of the region bounded by the graph of y = f(x), the x-axis, and the vertical lines x = a and x = b.

Fundamental Theorem of Calculus

A theorem that links the concepts of differentiating a function and integrating a function. The first part states if f is continuous, then d/dx ∫ₐˣ f(t)dt = f(x). The second part states if F is an antiderivative of f, then ∫ₐᵇ f(x)dx = F(b) - F(a).

Standard Integral Formulas (Must Memorize)

• ∫ xⁿ dx = (xⁿ⁺¹ / (n+1)) + C, for 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
• ∫ dx / √(a² - x²) = sin⁻¹(x/a) + C
• ∫ dx / (a² + x²) = (1/a)tan⁻¹(x/a) + C

Methods of Integration

Solving integrals often requires more than just applying standard formulas. There are three primary methods to tackle complex integrands.

1. Integration by Substitution: This method is used when the integrand is a composite function, specifically of the form f(g(x))g'(x). The core idea is to simplify the integral by substituting a part of the integrand with a new variable, say t. We choose t = g(x), which implies dt = g'(x)dx. This substitution transforms the original integral in terms of x into a simpler integral in terms of t. After integrating with respect to t, we substitute back t = g(x) to get the final answer in terms of x. For example, in ∫2x cos(x²) dx, we can substitute t = x², so dt = 2x dx, simplifying the integral to ∫cos(t) dt.

1. Integration by Partial Fractions: This technique applies to integrands that are rational functions, i.e., of the form P(x)/Q(x), where P(x) and Q(x) are polynomials and the degree of P(x) is less than the degree of Q(x). The method involves decomposing the rational function into a sum of simpler fractions. The form of the partial fractions depends on the factors of the denominator Q(x) (linear, repeated linear, quadratic, etc.). Once decomposed, each simpler fraction can be integrated using standard formulas.

1. Integration by Parts: This method is based on the product rule of differentiation and is used to integrate the product of two functions. The formula is: ∫u v dx = u ∫v dx - ∫(u' ∫v dx) dx. Here, u is the first function and v is the second function. The key to success is the correct choice of u and v. The LIATE rule helps in choosing the first function u: Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, Exponential. The function that comes first in this list should be chosen as u.

Integration by Parts: Applying the LIATE Rule

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Must Remember: Properties of Definite Integrals

• P₀: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt (Variable change doesn't alter value).
• P₁: ∫ₐᵇ f(x) dx = - ∫ₐᵇ f(x) dx (Interchanging limits changes the sign).
• P₂: ∫ₐᵃ f(x) dx = 0 (If limits are the same, the integral is zero).
• P₃: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx (Splitting the interval, where a < c < b).
• P₄ (King Property): ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b - x) dx. This is one of the most frequently used properties.
• P₅: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a - x) dx (A special case of P₄).
• P₆: ∫₀²ᵃ f(x) dx = ∫₀ᵃ f(x) dx + ∫₀ᵃ f(2a - x) dx.
• P₇ (Even/Odd Function Property): ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx, if f is an even function (f(-x) = f(x)).
• P₈ (Even/Odd Function Property): ∫₋ₐᵃ f(x) dx = 0, if f is an odd function (f(-x) = -f(x)).

Worked Mini-Examples

• {"example":"Evaluate ∫ tan⁴x sec²x dx","solution":"This is a case for substitution.\nLet u = tan x. Then du = sec²x dx.\nThe integral becomes ∫ u⁴ du.\n= u⁵/5 + C\nSubstituting back u = tan x, we get:\n= (tan⁵x)/5 + C"}
• {"example":"Evaluate ∫ x eˣ dx","solution":"This is a case for integration by parts (LIATE rule).\nLet u = x (Algebraic) and dv = eˣ dx (Exponential).\nThen du = dx and v = ∫eˣ dx = eˣ.\nUsing the formula ∫u dv = uv - ∫v du:\n∫x eˣ dx = x(eˣ) - ∫eˣ dx\n= x eˣ - eˣ + C\n= eˣ(x - 1) + C"}
• {"example":"Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx","solution":"Let I = ∫₀^(π/2) (sin x) / (sin x + cos x) dx ... (1)\nUsing property ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a - x) dx:\nI = ∫₀^(π/2) (sin(π/2 - x)) / (sin(π/2 - x) + cos(π/2 - x)) dx\nI = ∫₀^(π/2) (cos x) / (cos x + sin x) dx ... (2)\nAdding (1) and (2):\n2I = ∫₀^(π/2) (sin x + cos x) / (sin x + cos x) dx = ∫₀^(π/2) 1 dx\n2I = [x] from 0 to π/2 = π/2\nTherefore, I = π/4."}

Common Exam Traps

Don't forget '+C'! Forgetting the constant of integration C in indefinite integrals is a very common and costly mistake. Always add it as the final step.

LIATE Rule is a Guideline: While the LIATE rule is excellent, there are exceptions. The goal is to choose 'u' such that its derivative u' is simpler than u, and 'v' is easily integrable.

Definite Integral Properties: For definite integrals with symmetric limits like ∫₋ₐᵃ f(x) dx, always check if the function is even or odd first. It can save a lot of calculation time. If it's an odd function, the answer is immediately 0.

Substitution in Definite Integrals: When you use substitution in a definite integral, you must either change the limits of integration to the new variable 't' or substitute back to 'x' before applying the original limits. Don't mix the two.

Quick Revision Check

• What is the primary difference between an indefinite and a definite integral? An indefinite integral ∫f(x)dx gives a family of functions (F(x) + C), while a definite integral ∫ₐᵇ f(x)dx gives a single numerical value.
• What is the integral of sec x? ∫ sec x dx = ln|sec x + tan x| + C.
• If f(x) = x³cos(x), is ∫₋₂² f(x) dx equal to 0? Why? Yes. The function f(x) is an odd function because f(-x) = (-x)³cos(-x) = -x³cos(x) = -f(x). The definite integral of an odd function over a symmetric interval [-a, a] is always zero.
• Which method of integration would be best for ∫ (x+1) / (x²-4) dx? Integration by Partial Fractions, because the integrand is a rational function whose denominator can be factored as (x-2)(x+2).

Frequently Asked Questions about Integrals

Frequently Asked Questions