Class 9 Mathematics Chapter 7 Integrals: 30 MCQs with Complete Answers & Explanations

The revised CBSE Class 9 Mathematics curriculum (2024-25) has shifted away from purely computational problem-solving toward conceptual understanding and faster problem recognition. Multiple-choice questions (MCQs) are now the primary assessment tool in periodic tests, pre-board exams, and even board examinations. Why this change? MCQs force you to distinguish between correct and plausible-but-wrong answers—exactly what happens in real-world mathematics and competitive exams like JEE. For Chapter 7 (Integrals), MCQs test whether you can: (1) identify the antiderivative of a function instantly, (2) apply properties of definite integrals (linearity, additivity), (3) connect the Fundamental Theorem of Calculus to both derivative and integral concepts, and (4) avoid common algebra mistakes like forgetting constants of integration or misapplying limits. Each MCQ typically presents four options: one correct answer, one 'almost right' trap (e.g., missing a constant C), one conceptually wrong distractor, and one mathematically impossible choice. Mastering MCQs requires speed, precision, and deep conceptual clarity—not just formula memorization. This quiz builds all three skills progressively.

**Question 1:** The indefinite integral of 5x⁴ is: (A) x⁵ + C (B) 5x⁵ + C (C) 5x⁵/5 + C (D) 20x³ + C **Answer:** (A) x⁵ + C **Reason:** Apply the power rule: ∫5x⁴ dx = 5 · x⁵/5 + C = x⁵ + C. Trap: (B) forgets to divide by the new exponent. --- **Question 2:** Find ∫(3x² + 2x) dx: (A) 6x + 2 + C (B) x³ + x² + C (C) 3x³ + 2x² + C (D) x³ + 2x² + C **Answer:** (B) x³ + x² + C **Reason:** Integrate term-by-term: ∫3x² dx = x³, ∫2x dx = x², so answer is x³ + x² + C. Trap: (C) misses dividing 3x³ by 3. --- **Question 3:** If F'(x) = 7, then F(x) is: (A) 7x + C (B) 7 + C (C) 0 + C (D) 7x **Answer:** (A) 7x + C **Reason:** The antiderivative of constant 7 is 7x + C (where C is an arbitrary constant). Trap: (B) confuses derivative and integral. --- **Question 4:** ∫1 dx equals: (A) 1 (B) x + C (C) 0 + C (D) x **Answer:** (B) x + C **Reason:** The antiderivative of 1 is x + C. Always include the constant of integration in indefinite integrals. Trap: (A) forgets the variable and constant. --- **Question 5:** What is ∫(x³ − x) dx? (A) 3x² − 1 + C (B) x⁴/4 − x²/2 + C (C) x⁴ − x² + C (D) 4x³ − 1 + C **Answer:** (B) x⁴/4 − x²/2 + C **Reason:** Apply power rule separately: ∫x³ dx = x⁴/4 and ∫x dx = x²/2. Trap: (A) is the derivative, not the integral. --- **Question 6:** ∫e^x dx is: (A) e^(x+1) + C (B) e^x + C (C) x·e^x + C (D) e^x/x + C **Answer:** (B) e^x + C **Reason:** The integral of e^x is e^x itself (this is a memorized standard form). Trap: (A) treats e^x like a polynomial. --- **Question 7:** Find ∫(1/x) dx for x > 0: (A) −1/x² + C (B) ln(x) + C (C) 1 + C (D) x² + C **Answer:** (B) ln(x) + C **Reason:** The antiderivative of 1/x is the natural logarithm ln|x| + C (use positive form for x > 0). This is a standard integral. Trap: (D) is the antiderivative of 1/2x². --- **Question 8:** ∫(2sin(x)) dx equals: (A) 2cos(x) + C (B) −2cos(x) + C (C) 2sin(x) + C (D) sin²(x) + C **Answer:** (B) −2cos(x) + C **Reason:** ∫sin(x) dx = −cos(x) + C, so ∫2sin(x) dx = −2cos(x) + C. Trap: (A) forgets the negative sign. --- **Question 9:** ∫(cos(x)) dx is: (A) −sin(x) + C (B) sin(x) + C (C) cos²(x) + C (D) cos(x) + C **Answer:** (B) sin(x) + C **Reason:** The standard antiderivative of cos(x) is sin(x) + C. Trap: (A) reverses the sign and confuses this with ∫sin(x) dx. --- **Question 10:** What is the constant of integration and why is it necessary? (A) An arbitrary constant added to make the answer unique. (B) A symbol representing any constant whose derivative is zero. (C) Required only when limits are given in definite integrals. (D) A correction factor for algebraic errors. **Answer:** (B) A symbol representing any constant whose derivative is zero. **Reason:** Since d/dx(C) = 0 for any constant C, indefinite integrals must include ∫f(x) dx to show infinitely many antiderivatives. Trap: (A) is partially true but misses the mathematical reason.

**Question 11:** Evaluate the definite integral ∫₀² 3x dx: (A) 3 (B) 6 (C) 12 (D) 4 **Answer:** (B) 6 **Reason:** ∫₀² 3x dx = [3x²/2]₀² = 3(4)/2 − 0 = 6. Apply the Fundamental Theorem: evaluate the antiderivative at both limits and subtract. --- **Question 12:** What is ∫₁³ (x² + 1) dx? (A) 26/3 (B) 32/3 (C) 9 + 2 (D) 10 **Answer:** (B) 32/3 **Reason:** ∫₁³ (x² + 1) dx = [x³/3 + x]₁³ = (27/3 + 3) − (1/3 + 1) = 12 − 4/3 = 32/3. Trap: (A) incorrectly calculates the antiderivative evaluation. --- **Question 13:** Using the property ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx, if ∫₂⁵ f(x) dx = 10, what is ∫₅² f(x) dx? (A) 10 (B) −10 (C) 0 (D) 5 **Answer:** (B) −10 **Reason:** Reversing integration limits flips the sign: ∫₅² f(x) dx = −∫₂⁵ f(x) dx = −10. Trap: (A) ignores the limit reversal property. --- **Question 14:** If ∫₀¹ f(x) dx = 4 and ∫₁³ f(x) dx = 6, find ∫₀³ f(x) dx: (A) 2 (B) 10 (C) 24 (D) 1.5 **Answer:** (B) 10 **Reason:** By additivity of integrals: ∫₀³ f(x) dx = ∫₀¹ f(x) dx + ∫₁³ f(x) dx = 4 + 6 = 10. Trap: (A) subtracts instead of adds. --- **Question 15:** Evaluate ∫₀^(π/2) sin(x) dx: (A) 0 (B) 1 (C) π/2 (D) −1 **Answer:** (B) 1 **Reason:** ∫₀^(π/2) sin(x) dx = [−cos(x)]₀^(π/2) = −cos(π/2) − (−cos(0)) = 0 + 1 = 1. Trap: (A) confuses this with ∫₀^π sin(x) dx = 0. --- **Question 16:** What is ∫₁² (1/x) dx? (A) 0 (B) ln(2) (C) 1 − ln(2) (D) e − 1 **Answer:** (B) ln(2) **Reason:** ∫₁² (1/x) dx = [ln(x)]₁² = ln(2) − ln(1) = ln(2) − 0 = ln(2). Trap: (C) misapplies the antiderivative formula. --- **Question 17:** State the Fundamental Theorem of Calculus (Part 1): (A) d/dx[∫ₐˣ f(t) dt] = f(x) (B) ∫ₐᵇ f(x) dx = F(b) − F(a) for any F where F' = f (C) Both (A) and (B) (D) Neither; they are unrelated concepts. **Answer:** (C) Both (A) and (B) **Reason:** Both statements are correct: (A) is the derivative part, (B) is the evaluation part. Together, they form the Fundamental Theorem. Trap: (D) misses the connection between differentiation and integration. --- **Question 18:** If F(x) = ∫₁ˣ (2t + 1) dt, find F'(x): (A) 2x + 1 (B) x² + x (C) 2x + 1 + C (D) x² + x + C **Answer:** (A) 2x + 1 **Reason:** By the Fundamental Theorem, F'(x) = 2x + 1 (the integrand evaluated at x). Trap: (B) forgets to apply the theorem and computes the full antiderivative instead. --- **Question 19:** Evaluate ∫₋₂² x³ dx (where x³ is an odd function): (A) −16 (B) 0 (C) 16 (D) 8 **Answer:** (B) 0 **Reason:** The integral of an odd function over a symmetric interval [−a, a] always equals zero. Since x³ is odd, ∫₋₂² x³ dx = 0. Trap: (A) miscalculates without recognizing the symmetry property. --- **Question 20:** What is ∫₀¹ e^(2x) dx? (A) e² − 1 (B) (e² − 1)/2 (C) e² + 1 (D) 2(e − 1) **Answer:** (B) (e² − 1)/2 **Reason:** ∫e^(2x) dx = e^(2x)/2 + C. Evaluate: [e^(2x)/2]₀¹ = e²/2 − 1/2 = (e² − 1)/2. Trap: (A) forgets to divide by the coefficient 2 in the exponent.

**Question 21:** **Assertion (A):** The derivative of ∫ₐˣ f(t) dt with respect to x is f(x). **Reason (R):** This follows directly from the Fundamental Theorem of Calculus (Part 1). (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The Fundamental Theorem Part 1 explicitly states this relationship. The reason directly justifies the assertion. Trap: (B) ignores the causal link. --- **Question 22:** **Assertion (A):** ∫₀¹ f(x) dx + ∫₁² f(x) dx = ∫₀² f(x) dx for any continuous function f. **Reason (R):** This is because the integral is additive over adjacent intervals. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The additivity property is a fundamental axiom of the Riemann integral. R correctly names and explains A. Trap: (B) separates two inseparable concepts. --- **Question 23:** **Assertion (A):** ∫(x⁻¹) dx = ln|x| + C for all non-zero x. **Reason (R):** Because the derivative of ln|x| with respect to x is 1/x. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** d/dx[ln|x|] = 1/x, which confirms the antiderivative. R is the proof of A. Trap: (C) denies R incorrectly. --- **Question 24:** **Assertion (A):** ∫₋₃³ (x² + 3x) dx = ∫₋₃³ x² dx (because the x term is odd and cancels). **Reason (R):** Even functions integrate symmetrically; odd function integrals over symmetric intervals equal zero. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** x² is even (integrates normally), 3x is odd (integrates to zero over [−3,3]), so A follows from R. Trap: (B) misses the logical flow. --- **Question 25:** **Assertion (A):** If ∫₁⁵ f(x) dx = 12 and f is continuous, then there exists some c ∈ [1, 5] where f(c) = 12/(5−1) = 3. **Reason (R):** This is a direct consequence of the Mean Value Theorem for Integrals. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The MVTI states: ∫ₐᵇ f(x) dx = f(c)(b − a) for some c ∈ [a,b]. This is exactly A. Trap: (D) denies R incorrectly. --- **Question 26:** **Assertion (A):** ∫₂⁵ x dx ≠ ∫₅² x dx **Reason (R):** Reversing limits introduces a negative sign: ∫₅² x dx = −∫₂⁵ x dx. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** ∫₂⁵ x dx = 25/2 − 2 = 21/2, while ∫₅² x dx = −21/2. R is the theorem justifying this difference. Trap: (B) ignores the causal relationship. --- **Question 27:** **Assertion (A):** The constant of integration C is unnecessary when evaluating a definite integral ∫ₐᵇ f(x) dx. **Reason (R):** The antiderivative constants cancel when computing F(b) − F(a). (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** [F(x) + C]ₐᵇ = (F(b) + C) − (F(a) + C) = F(b) − F(a). The C terms cancel, so C is irrelevant in definite integrals. R explains why A is true. Trap: (B) severs the logical connection. --- **Question 28:** **Assertion (A):** ∫₀^(π) sin(x) dx ≠ ∫₀^(π/2) sin(x) dx. **Reason (R):** Because the integrand sin(x) is positive on [0, π/2] but oscillates sign differently on [0, π]. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (B) Both A and R are true, but R is not the correct explanation of A. **Reason:** A is true: ∫₀^(π) sin(x) dx = 2, while ∫₀^(π/2) sin(x) dx = 1. But R's explanation is imprecise—sin is positive on entire [0, π/2], not oscillating. The difference is simply that [0, π] is a larger interval. Trap: (A) overlooks the flawed reasoning. --- **Question 29:** **Assertion (A):** d/dx[∫₁ˣ (t² + 2t) dt] = x² + 2x. **Reason (R):** By the Fundamental Theorem, the derivative of an integral (with variable upper limit) equals the integrand evaluated at that limit. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** Apply FTC Part 1: d/dx[∫₁ˣ (t² + 2t) dt] = x² + 2x (evaluating the integrand at x). R is the precise statement of why A holds. Trap: (B) ignores the direct logical chain. --- **Question 30:** **Assertion (A):** ∫₀² (x² − 2x) dx = ∫₀² x(x − 2) dx (both integrals are equal). **Reason (R):** Because algebraic factorization does not change the value of a definite integral. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** x² − 2x = x(x − 2) is an identity, so integrating either form yields the same result: [x³/3 − x²]₀² = 8/3 − 4 = −4/3. R correctly explains why. Trap: (B) misses the logical necessity of the equivalence.

**Trap 1: Forgetting the Constant of Integration (C)** In indefinite integrals, always include + C. ∫3x² dx = x³ + C, NOT x³. Without C, you're saying there's only one antiderivative, which is mathematically incorrect. Board examiners explicitly check for this. **Trap 2: Misapplying the Power Rule** ∫xⁿ dx = x^(n+1)/(n+1) + C, not x^(n+1)/n + C. The denominator is the *new* exponent (n+1), not the old one (n). A common error: ∫5x⁴ dx → many students write 5x⁵/4 instead of x⁵ + C. **Trap 3: Confusing Derivative and Integral** If d/dx[F(x)] = f(x), then ∫f(x) dx = F(x) + C. The operations are inverse. For example: ∫6x dx ≠ 6, but d/dx[3x²] = 6x. Watch MCQ options that show the derivative (like "6x + 2") when the question asks for the integral. **Trap 4: Forgetting the Chain Rule Adjustment for Composite Functions** ∫e^(2x) dx = e^(2x)/2 + C, not e^(2x) + C. When the inner function is linear (like 2x), divide by its coefficient. Many students skip this division. **Trap 5: Incorrect Limit Substitution in Definite Integrals** Always evaluate F(upper) − F(lower), not F(lower) − F(upper). And be careful with signs. For ∫₀² 3x dx = [3x²/2]₀² = 3(4)/2 − 3(0)/2 = 6, not negative. **Trap 6: Reversing Integral Limits Without Changing Sign** ∫₅² f(x) dx = −∫₂⁵ f(x) dx (the sign flips). If an MCQ says ∫₂⁵ f(x) dx = 10, then ∫₅² f(x) dx = −10, not 10. **Trap 7: Ignoring Even/Odd Function Symmetry** ∫₋ₐᵃ (odd function) dx = 0 always. ∫₋ₐᵃ (even function) dx = 2∫₀ᵃ (even function) dx. Many students compute these integrals numerically when they could shortcut using symmetry. Example: ∫₋₃³ x³ dx = 0 instantly (no calculation needed). **Trap 8: Misunderstanding the Fundamental Theorem** d/dx[∫ₐˣ f(t) dt] = f(x), NOT f(x) + C. There's no constant here because the limits are specific. The constant only appears in indefinite integrals. **Trap 9: Forgetting Absolute Value in Logarithmic Integrals** ∫(1/x) dx = ln|x| + C for all x ≠ 0 (use absolute value to handle both positive and negative x). Students often write ln(x) + C, forgetting the domain extends to negative x. **Trap 10: Misapplying Linearity—Dividing Instead of Multiplying** ∫(5f(x)) dx = 5∫f(x) dx (multiply by the constant). However, ∫(f(x)/5) dx = (1/5)∫f(x) dx. Be careful with division: it's a scalar coefficient, so it factors out normally. Trap MCQs sometimes show divided-out answers that are technically wrong in the context of the original integrand.

CBSE Class 9 periodic tests and exams typically allocate 2–3 hours for 40–50 marks of Mathematics, which includes multiple sections. Here's how to pace yourself on Integrals MCQs: **Pre-Exam Preparation (1–2 weeks before):** 1. Master all 10 easy MCQs first (30–45 seconds each). These build confidence and ensure you capture 10/10 points if they appear on the exam. 2. Drill medium MCQs daily (1–2 minutes each). Use a timer. Compute ∫₀¹ 3x dx in 60 seconds; mark answers without showing all algebra on rough work—just the final substitution. 3. Study assertion–reason MCQs by theme (e.g., all on the Fundamental Theorem together). Assertion–reason requires 1.5–2 minutes because you must evaluate both statements. **During the Exam:** - **First Pass (5 minutes):** Skim all MCQs. Answer the 4–5 you're 100% sure of instantly (typically easy MCQs 1–5). - **Second Pass (8–10 minutes):** Tackle medium MCQs 11–20. If you can identify the antiderivative in your head, write it down and evaluate the limits. Don't over-think. - **Third Pass (5–7 minutes):** Approach assertion–reason MCQs (21–30). Read both statements carefully. If both are true, decide whether R explains A. Use elimination: cross off options where A is obviously false first. - **Buffer (2–3 minutes):** Review your marked answers; double-check signs in definite integrals and constants in indefinite integrals. **Speed Hacks for Common Questions:** - **Antiderivatives:** Memorize the 5–6 standard forms (xⁿ, e^x, sin x, cos x, 1/x, 1/(1+x²)). Recognize them instantly. - **Definite Integrals with Odd/Even Functions:** Don't calculate—use symmetry directly. Saves 30 seconds per question. - **Limit Reversal:** Check if the MCQ reverses limits. If yes, flip the sign. One-second check. - **Chain Rule in Exponents:** If you see ∫e^(ax) dx, immediately think e^(ax)/a + C. No derivation needed in a timed MCQ. **Common Timing Pitfalls to Avoid:** - Don't recalculate ∫f(x) dx from scratch for every definite integral. Compute the antiderivative once, then substitute limits. - Avoid writing out the full Riemann sum definition or historical context—straight to the answer. - Skip double-checking arithmetic unless the answer seems wrong. Move forward. **Post-Exam Reflection:** - If you got a medium MCQ wrong, identify: Did you forget C? Misapply the power rule? Reverse limits by mistake? Log this pattern and drill 3–4 similar problems before the next exam. - If assertion–reason tripped you up, spend 10 minutes on that day practicing *why* R explains A (or why it doesn't). Start a 3-day free trial at cbsetutor.ai to access full video solutions for every MCQ, step-by-step reasoning, and personalized quizzes tailored to your weak areas in Chapter 7.