∫[a, b] f(x) dx equals:

−∫[b, a] f(x) dx. Interchanging limits changes the sign.

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 1 Chapter 6: Definite Integration

Short notes, key concepts, quick revision and practice MCQs with answers

A definite integral has limits and gives a number; here ∫[a, b] f(x) dx means the integral of f(x) from x = a to x = b. The chapter states the fundamental theorem of integral calculus, evaluates definite integrals using the methods of indefinite integration and applies the properties of definite integrals to simplify them.

Key concepts: Definite Integration

Definite integral: ∫[a, b] f(x) dx is the definite integral of f from a (lower limit) to b (upper limit). Its value is a number; no constant of integration is needed.
Fundamental theorem: If ∫f(x) dx = F(x) + c, then ∫[a, b] f(x) dx = [F(x)] from a to b = F(b) − F(a).
Evaluation by substitution: When substituting t = g(x), change the limits too: x = a gives t = g(a) and x = b gives t = g(b).
Property 1 and 2: ∫[a, a] f(x) dx = 0 and ∫[a, b] f(x) dx = −∫[b, a] f(x) dx. Also ∫[a, b] f(x) dx = ∫[a, b] f(t) dt.
Property 3: ∫[a, b] f(x) dx = ∫[a, c] f(x) dx + ∫[c, b] f(x) dx for a < c < b.
Property 4 (a − x): ∫[0, a] f(x) dx = ∫[0, a] f(a − x) dx and ∫[a, b] f(x) dx = ∫[a, b] f(a + b − x) dx.
Even and odd functions: ∫[−a, a] f(x) dx = 2∫[0, a] f(x) dx if f is even (f(−x) = f(x)) and = 0 if f is odd (f(−x) = −f(x)).

Quick revision: important points

∫[0, 1] x dx = 1/2; ∫[0, 1] eˣ dx = e − 1; ∫[1, e] (1/x) dx = 1.
x², x⁴ are even; x, x³ are odd functions.
Always change limits when substituting.
Interchanging limits changes the sign of the integral.

Practice MCQs with answers

∫[0, 1] x dx equals:
1. 1
2. 1/2
3. 0
4. 2
Show answer
Answer: B. 1/2
[x²/2]₀¹ = 1/2.
∫[1, 2] 2x dx equals:
1. 2
2. 3
3. 4
4. 1
Show answer
Answer: B. 3
[x²]₁² = 4 − 1 = 3.
∫[0, 2] x² dx equals:
1. 4
2. 8/3
3. 8
4. 2/3
Show answer
Answer: B. 8/3
[x³/3]₀² = 8/3.
∫[a, a] f(x) dx equals:
1. 2a
2. f(a)
3. 0
4. 1
Show answer
Answer: C. 0
Equal limits give zero.
∫[a, b] f(x) dx equals:
1. ∫[b, a] f(x) dx
2. −∫[b, a] f(x) dx
3. 2∫[0, b] f(x) dx
4. 0
Show answer
Answer: B. −∫[b, a] f(x) dx
Interchanging limits changes the sign.

All Class 12 Maths and Statistics chapters

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