Integrals – Class 12 Mathematics notes, important questions and MCQs

Chapter overview

Integration is the reverse of differentiation. Indefinite integrals give families of antiderivatives; definite integrals give net area and are evaluated by the fundamental theorem.

Methods

• Standard forms: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1); ∫1/x dx = log|x| + C; ∫eˣ dx = eˣ + C.
• Substitution, partial fractions and integration by parts: ∫u v dx = u∫v dx − ∫(u′∫v dx) dx (ILATE order).
• Special forms: ∫dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C; ∫dx/√(a² − x²) = sin⁻¹(x/a) + C.

Definite integrals

• ∫ₐᵇ f(x) dx = F(b) − F(a).
• ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx.
• ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even, and 0 if f is odd.

Key terms

Antiderivative

function whose derivative is the given function

ILATE

order to choose u in integration by parts: Inverse, Log, Algebraic, Trig, Exponential

Partial fractions

splitting a rational function into simpler fractions so that each can be integrated

Formulas

• ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C
• ∫eˣ[f(x) + f′(x)] dx = eˣf(x) + C
• ∫₋ₐᵃ odd f = 0

Important questions with answers

Quick revision

• Don't forget + C for indefinite integrals.
• Odd function over symmetric limits gives 0.
• ILATE for parts.

Exam tips

• Use the property ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx for sin/cos symmetric integrals.
• For ∫ f′(x)/f(x) dx the answer is log|f(x)| + C — check whether the numerator is the derivative of the denominator.

Practice MCQs

1. ∫ cos x dx equals

   1. sin x + C
   2. −sin x + C
   3. cos x + C
   4. −cos x + C
   Show answer

   Answer: (a) sin x + C. Reverse of derivative.

2. ∫ 1/x dx equals

   1. x² + C
   2. 1/x² + C
   3. log|x| + C
   4. eˣ + C
   Show answer

   Answer: (c) log|x| + C. Standard form.

3. ∫₀¹ x² dx equals

   1. 1/2
   2. 1/3
   3. 1
   4. 2/3
   Show answer

   Answer: (b) 1/3. [x³/3]₀¹.

4. ∫₋₁¹ x³ dx equals

   1. 1/2
   2. 1/4
   3. 2
   4. 0
   Show answer

   Answer: (d) 0. Odd function.

5. ∫ sec²x/tan x dx equals

   1. log|tan x| + C
   2. log|sec x| + C
   3. tan²x/2 + C
   4. sec x + C
   Show answer

   Answer: (a) log|tan x| + C. f′/f form.

6. ∫ x eˣ dx equals

   1. x eˣ + C
   2. eˣ(x − 1) + C
   3. eˣ(x + 1) + C
   4. x² eˣ/2 + C
   Show answer

   Answer: (b) eˣ(x − 1) + C. By parts.

7. ∫ dx/(x² + 9) equals

   1. 3 tan⁻¹(3x) + C
   2. tan⁻¹(x/3) + C
   3. (1/9) tan⁻¹ x + C
   4. (1/3) tan⁻¹(x/3) + C
   Show answer

   Answer: (d) (1/3) tan⁻¹(x/3) + C. a = 3.

8. ∫₀^(π/2) sin x/(sin x + cos x) dx equals

   1. π/2
   2. 1
   3. π/4
   4. 0
   Show answer

   Answer: (c) π/4. Use f(a − x) property: 2I = π/2.