1. How to Use This Page
This page contains complete exercise solutions for Chapter 11 — Integral Calculus of the Tamil Nadu State Board Class 11 Mathematics textbook. Solve each exercise independently first, then verify your answers and methods below. All worked examples include full step-by-step numerical solutions in code blocks.
2. Key Concepts (Quick Recap)
Standard Integration Formulas
| Integral | Result | ||
|---|---|---|---|
| ∫xⁿ dx | x^(n+1)/(n+1) + C, 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 | ||
| ∫csc²x dx | −cot x + C | ||
| ∫sec x tan x dx | sec x + C | ||
| ∫csc x cot x dx | −csc x + C | ||
| ∫dx/√(1−x²) | sin⁻¹x + C | ||
| ∫dx/(1+x²) | tan⁻¹x + C | ||
| ∫1 dx | x + C | ||
| ∫0 dx | C |
Key Properties
Methods of Integration
1. Direct integration — Use standard formulas 2. Substitution — Let u = g(x), transform to u-variable 3. Partial fractions — Decompose rational functions 4. Integration by parts — ∫u dv = uv − ∫v du (for Class 12)
3. I. Choose the Best Answer (MCQs)
Q1. ∫(2x + 3) dx equals:
(a) x² + 3x + C   (b) 2x² + 3x + C   (c) x² + C   (d) 2x + C
Q2. ∫₀¹ x² dx equals:
(a) 1   (b) 1/2   (c) 1/3   (d) 1/4
Q3. ∫(1/x) dx equals:
| (a) x⁻¹ + C   (b) ln x + C   (c) ln\ | x\ | + C   (d) x + C |
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| Answer: (c) ln\ | x\ | + C |
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| Explanation: The power rule does not apply when n = −1. Instead, ∫(1/x) dx = ln\ | x\ | + C. The absolute value is needed because ln is not defined for negative numbers. |
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Q4. If d/dx[F(x)] = f(x), then ∫f(x) dx equals:
(a) F(x) + C   (b) f(x) + C   (c) F'(x) + C   (d) f'(x) + C
Q5. ∫sin(2x) dx equals:
(a) cos(2x) + C   (b) −cos(2x) + C   (c) −(1/2)cos(2x) + C   (d) (1/2)cos(2x) + C
4. II. Fill in the Blanks
1. ∫5 dx = 5x + C.
2. ∫x⁻² dx = −x⁻¹ + C = −1/x + C.
3. ∫cos x dx = sin x + C.
4. ∫₀² 3x² dx = 8. (3·2³/3 = 8)
5. The constant C in indefinite integrals is called the constant of integration.
6. ∫eˣ dx = eˣ + C.
7. ∫(sec²x + csc²x) dx = tan x − cot x + C.
8. If ∫f(x) dx = F(x) + C, then ∫ₐᵇ f(x) dx = F(b) − F(a).
9. ∫(1 + x + x²) dx = x + x²/2 + x³/3 + C.
10. ∫sin(3x) dx = −(1/3)cos(3x) + C.
11. ∫dx/(1 + x²) = tan⁻¹x + C.
12. ∫₂³ dx = 1 (integral of 1 over interval of length 1).
5. III. True or False (with Corrections)
1. Statement: ∫x⁻¹ dx = x⁰/0 + C.
| Answer: False. The power rule ∫xⁿ dx = x^(n+1)/(n+1) does not apply for n = −1. The correct result is ln\ | x\ | + C. |
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2. Statement: The integral of a constant k is kx + C. Answer: True. ∫k dx = kx + C.
3. Statement: ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx. Answer: True. Reversing the limits changes the sign.
4. Statement: ∫sin x dx = cos x + C. Answer: False. ∫sin x dx = −cos x + C. The derivative of cos x is −sin x, so the integral of sin x is −cos x.
5. Statement: Two different functions can have the same derivative. Answer: True. They differ by a constant. If F'(x) = f(x), then (F(x) + C)' = f(x) for any constant C.
6. IV. Match the Following
| Column A | Column B | ||
|---|---|---|---|
| 1. ∫x³ dx | a. ln\ | x\ | + C |
| 2. ∫(1/x) dx | b. x + C | ||
| 3. ∫5 dx | c. x⁴/4 + C | ||
| 4. ∫cos x dx | d. −cos x + C | ||
| 5. ∫sin x dx | e. 5x + C | ||
| 6. ∫1 dx | f. sin x + C |
7. V. Answer Briefly (2-Mark Questions)
Q1. Evaluate ∫(3x² + 4x − 7) dx.
Q2. Evaluate ∫₀¹ (2x + 1) dx.
Q3. Evaluate ∫(sin x + cos x) dx.
Q4. Find ∫(1/√x) dx.
8. VI. Answer in Detail (5-Mark Questions)
Q1. Evaluate ∫(2x + 3)/(x² + 3x + 5) dx using substitution.
Q2. Evaluate ∫₀² (3x² − 2x + 1) dx.
Q3. Evaluate ∫sin²x dx using the identity sin²x = (1 − cos 2x)/2.
Q4. Evaluate ∫x√(x + 2) dx using substitution.
Q5. Evaluate ∫₀^(π/2) cos²x dx using the identity cos²x = (1 + cos 2x)/2.
9. Common Mistakes to Avoid
1. Forgetting the constant of integration (+C): Every indefinite integral must include +C. Without it, the answer is incomplete and marks are lost.
| 2. Applying power rule for n = −1: ∫(1/x) dx = ln\ | x\ | + C, NOT x⁰/0. The power rule formula ∫xⁿ dx = x^(n+1)/(n+1) + C only works for n ≠ −1. |
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3. Sign errors in trigonometric integrals: ∫sin x dx = −cos x + C (there IS a minus sign). ∫csc²x dx = −cot x + C. Always verify by differentiating your answer.
4. Wrong substitution choice: Choose u so that du appears in the integrand (or a constant multiple of it). If nothing simplifies after substitution, try a different u.
5. Forgetting to substitute back: After integrating in terms of u, you must convert back to x before writing the final answer (for indefinite integrals).
6. Definite integral limits: When using substitution for a definite integral, either change the limits to u-values or substitute back to x before evaluating. Don't mix x-limits with u-expressions.
10. Exam Checklist
- [ ] Know all standard integration formulas (power, trig, exponential, logarithmic)
- [ ] Can integrate polynomials term by term
- [ ] Can apply substitution method: identify u, compute du, transform, integrate, back-substitute
- [ ] Remember ∫(1/x) dx = ln\|x\| + C (exception to power rule)
- [ ] Can evaluate definite integrals using the Fundamental Theorem
- [ ] Know properties: ∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx, ∫ₐᵇ = ∫ₐᶜ + ∫ᶜᵇ
- [ ] Can use trig identities: sin²x = (1−cos2x)/2, cos²x = (1+cos2x)/2
- [ ] Always include +C for indefinite integrals
- [ ] Can verify answers by differentiating the antiderivative
- [ ] Know ∫₀^(π/2) sinⁿx dx and ∫₀^(π/2) cosⁿx dx patterns for small n