Integrals Class 12 Maths Chapter 7 Notes

Key Terms in Integration

Integration

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

Indefinite Integral

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

Definite Integral

An integral with upper and lower limits, denoted by ∫ₐᵇ f(x) dx. It represents the algebraic sum of areas of the region bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b.

Integrand

The function f(x) that is to be integrated.

Constant of Integration (C)

An arbitrary constant added to the result of an indefinite integral, representing the family of functions that have the same derivative.

Limits of Integration

The values 'a' (lower limit) and 'b' (upper limit) in a definite integral ∫ₐᵇ f(x) dx.

Fundamental Theorem of Calculus

The theorem that links differentiation and integration. If F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) - F(a).

Standard Integration Formulas (Formula Sheet)

Methods of Integration

When the integrand is not in a standard form, we use various methods to simplify it. The three primary methods are:

1. Integration by Substitution: This method is used when the integrand is a composite function, specifically of the form f(g(x)) * g'(x). We substitute t = g(x), which implies dt = g'(x) dx. The integral transforms into ∫ f(t) dt, which is often easier to solve. The key is to correctly identify the function g(x) whose derivative g'(x) is also present in the integral.

1. Integration by Partial Fractions: This method applies to rational functions P(x)/Q(x) where the degree of P(x) is less than the degree of Q(x). The first step is to factorize the denominator Q(x). Then, the rational function is decomposed into a sum of simpler fractions. For example, a term (px+q)/((x-a)(x-b)) can be written as A/(x-a) + B/(x-b). We find the values of A and B and then integrate the simpler fractions separately.

1. Integration by Parts: This method is used to integrate the product of two functions. It is based on the product rule of differentiation. The formula is: **∫ u v dx = u ∫ v dx - ∫ (u' * (∫ v dx)) dx. The choice of the first function (u) and the second function (v) is crucial. We use the LIATE** rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential) to decide which function to choose as u. The function that comes first in LIATE is chosen as u.

Applying Integration by Parts (LIATE Rule)

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Key Properties of Definite Integrals

• {"point":"P₀: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt (Changing the variable does not change the value)."}
• {"point":"P₁: ∫ₐᵇ f(x) dx = - ∫ₐᵇ f(x) dx (Swapping limits changes the sign)."}
• {"point":"P₂: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ₐᵇ f(x) dx, where a < c < b (Splitting the interval)."}
• {"point":"P₃ (King's Property): ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx. This is extremely useful."}
• {"point":"P₄: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx (A special case of P₃)."}
• {"point":"P₅: ∫₀²ᵃ f(x) dx = ∫₀ᵃ f(x) dx + ∫₀ᵃ f(2a-x) dx."}
• {"point":"P₆: ∫₀²ᵃ f(x) dx = 2∫₀ᵃ f(x) dx, if f(2a-x) = f(x). It equals 0, if f(2a-x) = -f(x)."}
• {"point":"P₇: ∫₋ᵃᵃ f(x) dx = 2∫₀ᵃ f(x) dx, if f(x) is an even function (f(-x) = f(x))."}
• {"point":"P₇ (cont.): ∫₋ᵃᵃ f(x) dx = 0, if f(x) is an odd function (f(-x) = -f(x))."}

Worked Examples

• {"heading":"Example 1: Integration by Substitution","problem":"Evaluate ∫ 2x sin(x²) dx","solution":"Let t = x². Then dt = 2x dx.\nThe integral becomes ∫ sin(t) dt = -cos(t) + C.\nSubstituting back t = x², the answer is -cos(x²) + C."}
• {"heading":"Example 2: Integration by Parts","problem":"Evaluate ∫ x eˣ dx","solution":"Using LIATE, 'x' is Algebraic (A) and 'eˣ' is Exponential (E). So, u = x and v = eˣ.\nu' = 1, ∫v dx = eˣ.\nUsing the formula: ∫ x eˣ dx = x(eˣ) - ∫ 1 * eˣ dx = x eˣ - eˣ + C = eˣ(x-1) + C."}
• {"heading":"Example 3: Using Properties of Definite Integrals","problem":"Evaluate ∫₋π/₂ π/₂ sin³(x) dx","solution":"Let f(x) = sin³(x). Check if it's odd or even.\nf(-x) = sin³(-x) = (-sin(x))³ = -sin³(x) = -f(x).\nSince f(x) is an odd function, by property P₇ (∫₋ᵃᵃ f(x) dx = 0 for odd f(x)), the value of the integral is 0."}

Board Exam Traps

Don't forget '+C': Forgetting the constant of integration in indefinite integrals is a very common mistake and leads to an immediate loss of marks. Always add '+C' at the end.

LIATE Rule: When using Integration by Parts, choosing the wrong function as 'u' can make the problem much harder or even unsolvable. Strictly follow the LIATE rule.

Sign Errors in Definite Integrals: Be extremely careful with signs when applying the Fundamental Theorem of Calculus: F(b) - F(a). A simple mistake like -(-value) becoming -value can cost you the entire question.

Modulus Functions: For integrals involving modulus, like ∫₋₁² |x| dx, always split the integral at the point where the expression inside the modulus becomes zero. Here, split at x=0: ∫₋₁⁰ (-x) dx + ∫₀² (x) dx.

Practice Questions with Solutions

• Q: What is the integral of 1 / (x²+a²)? A: (1/a) tan⁻¹(x/a) + C. This is a standard formula you must remember.
• Q: Which method would you use for ∫ (x+1)/(x²-5x+6) dx? A: Integration by Partial Fractions, because the integrand is a rational function and the denominator can be factored into (x-2)(x-3).
• Q: What is the value of ∫₀^(π/2) (sin x / (sin x + cos x)) dx using properties? A: π/4. Use the property ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx. Call the integral I, apply the property to get another I, add them (2I), and solve.
• Q: Differentiate eˣ(x-1). What does this tell you about ∫ eˣ(x-1) dx? A: The derivative is eˣ(1) + eˣ(x-1) = eˣ + xeˣ - eˣ = xeˣ. This confirms that ∫ xeˣ dx = eˣ(x-1) + C, verifying the result from Integration by Parts.

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