How to Use This Page
These are complete exercise solutions for Samacheer Kalvi 12th Business Maths — Integral Calculus I (Chapter 2). Solve each question yourself first, then verify your method and answer below. Pay special attention to the step-by-step methods — integration techniques are the backbone of Chapters 2 and 3.
Key Concepts (Quick Recap)
- Indefinite integral: ∫f(x) dx = F(x) + C, where F'(x) = f(x). Always include the constant of integration C.
- Power rule: ∫xⁿ dx = x^(n+1)/(n+1) + C, n ≠ −1
- Reciprocal rule: ∫(1/x) dx = ln|x| + C
- Exponential rule: ∫eˣ dx = eˣ + C; ∫aˣ dx = aˣ/ln a + C
- Standard forms: ∫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
- Substitution method: If integral contains f(g(x)) · g'(x), substitute u = g(x), du = g'(x) dx.
- Integration by parts: ∫u dv = uv − ∫v du. Choose u by LIATE priority: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
- Partial fractions: Decompose rational functions (P(x)/Q(x)) into simpler fractions based on factor types of Q(x).
- Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is the antiderivative.
- Properties of definite integrals:
- ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx
- ∫ₐᵃ f(x) dx = 0
- ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx
- ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx
I. Choose the Best Answer (MCQs)
Q1. ∫(1/√x) dx equals: (a) 2√x + C ✓ (b) √x + C (c) (1/2)√x + C (d) x^(1/2) + C
Q2. ∫₀¹ x(1−x)⁴ dx equals: (a) 1/5 (b) 1/30 ✓ (c) 1/6 (d) 1/120
Q3. ∫ sin x / cos²x dx equals: (a) sec x + C ✓ (b) cosec x + C (c) tan x + C (d) −sec x + C
Q4. In integration by parts ∫x eˣ dx, we choose: (a) u = eˣ, dv = x dx (b) u = x, dv = eˣ dx ✓ (c) u = x eˣ, dv = dx (d) u = 1, dv = x eˣ dx
Q5. ∫₀^(π/2) sin²x dx equals: (a) π/2 (b) π/4 ✓ (c) π (d) 0
Q6. ∫ dx / (x² + 4) equals: (a) (1/2) arctan(x/2) + C ✓ (b) arctan(x/2) + C (c) (1/2) arctan(2x) + C (d) arctan(2x) + C
Q7. ∫ eˣ(1 + x) dx equals: (a) x eˣ + C ✓ (b) eˣ + C (c) (1+x) eˣ + C (d) eˣ/x + C
(a) 0 (b) 2 (c) 1 ✓ (d) 4
| **Q8. ∫₀² | x−1 | dx equals:** |
|---|
Q9. The value of ∫₋₁¹ x³ dx is: (a) 1 (b) 0 ✓ (c) 2 (d) −1
Q10. Partial fraction decomposition of (2x+1)/((x+1)(x+2)) is: (a) 3/(x+1) − 1/(x+2) ✓ (b) 1/(x+1) + 1/(x+2) (c) 2/(x+1) + 1/(x+2) (d) −1/(x+1) + 3/(x+2)
II. Fill in the Blanks
| 1. ∫(1/x) dx = ______. **ln | x | + C** |
|---|
2. ∫eˣ dx = ______. eˣ + C
3. ∫₀ᵃ f(x) dx = ∫₀ᵃ f(______) dx. a − x
4. ∫ₐᵇ f(x) dx + ∫ᵇᶜ f(x) dx = ∫______ f(x) dx. ₐᶜ
5. ∫ sec²x dx = ______. tan x + C
6. In integration by parts, ∫u dv = ______. uv − ∫v du
7. ∫₀^(π/2) sin x dx = ______. 1
8. ∫ dx/(x² + a²) = ______. (1/a) arctan(x/a) + C
III. True or False
1. ∫₀ᵃ f(x) dx = −∫₀ᵃ f(a−x) dx → False. The correct property is ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx. There is no negative sign.
2. The constant of integration C is necessary in indefinite integrals but not in definite integrals. → True. Definite integrals give a numerical value; the constant cancels out. Indefinite integrals represent a family of curves, so C is essential.
3. ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx if f is an even function. → True. For an even function, f(−x) = f(x), so the integral over [−a, a] equals twice the integral over [0, a].
4. Integration by parts can be applied to any integral. → False. Integration by parts applies specifically to products of two functions. For simple integrands like ∫sin x dx or ∫eˣ dx, direct integration is more appropriate.
| 5. **∫ dx/(x² − a²) = (1/2a) ln | (x−a)/(x+a) | + C → True.** This is the standard result for the integral of 1/(x² − a²), derived using partial fractions. |
|---|
6. If f(x) ≤ g(x) on [a,b], then ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx. → True. This is the comparison property of definite integrals.
IV. Match the Following
| Left | Right | ||
|---|---|---|---|
| 1. ∫xⁿ dx | d. x^(n+1)/(n+1) + C | ||
| 2. ∫(1/x) dx | a. ln | x | + C |
| 3. ∫sec x tan x dx | e. sec x + C | ||
| 4. ∫₀^(π/2) sin²x dx | b. π/4 | ||
| 5. ∫eˣ(sin x + cos x) dx | c. eˣ sin x + C |
V. Answer Briefly
VI. Answer in Detail
Common Mistakes to Avoid
1. Forgetting the constant C in indefinite integrals: Always add "+ C" to indefinite integral answers. Examiners deduct marks for missing C.
2. Wrong choice of u in integration by parts: Follow the LIATE rule strictly. Choosing u incorrectly leads to more complex integrals, not simpler ones. Common error: choosing eˣ as u when integrating x eˣ.
3. Incorrect limits after substitution: When using substitution in definite integrals, you must change the limits to match the new variable. Example: if u = sin x, when x goes from 0 to π/2, u goes from 0 to 1.
4. Sign errors in definite integral evaluation: F(b) − F(a), not F(a) − F(b). Always substitute the upper limit first, then subtract the lower limit.
5. Incomplete partial fraction decomposition: For repeated linear factors like (x+1)², you must write A/(x+1) + B/(x+1)², not A/(x+1) + B/(x+1). Similarly, irreducible quadratic factors need (Ax+B)/(x²+px+q) form.
6. Misapplying the property ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx: This only works for limits 0 to a. It does not apply for arbitrary limits or when the integrand is not in a suitable form.
Exam Checklist
- [ ] I can apply standard integration formulas (power, exponential, trigonometric)
- [ ] I can use the substitution method correctly, including changing limits for definite integrals
- [ ] I can apply integration by parts using the LIATE rule
- [ ] I can decompose rational functions into partial fractions and integrate
- [ ] I can evaluate definite integrals and apply their properties
- [ ] I can use ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx to simplify integrals
- [ ] I can handle integrals involving absolute values by splitting the domain
- [ ] I remember to add "+ C" for indefinite integrals and never for definite integrals