Integrals — Class 12 MCQs with Answers

Integrals — important questions & MCQs with answers (Class 12 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   ∫ x² dx =

   A.x³ + C

   B.x³/3 + C✓ Correct

   C.2x + C

   D.3x² + C

   Answer: B. x³/3 + C

   Explanation: Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Here n=2, so x³/3 + C — the tempting mistake is skipping the ÷3, which gives just x³.

2. Q2Easy

   ∫ xⁿ dx (n ≠ −1) =

   A.xⁿ + C

   B.xⁿ⁺¹

   C.xⁿ⁺¹/(n + 1) + C✓ Correct

   D.nxⁿ⁻¹

   Answer: C. xⁿ⁺¹/(n + 1) + C

   Explanation: Power rule: raise the exponent by 1 and divide by the new exponent, xⁿ⁺¹/(n+1) + C. Forgetting the ÷(n+1) — just writing xⁿ⁺¹ — is the most common slip.

3. Q3Easy

   ∫ u · dv = ?

   A.uv + ∫ v · du

   B.u · v

   C.∫ u · ∫ v

   D.uv − ∫ v · du✓ Correct

   Answer: D. uv − ∫ v · du

   Explanation: Integration by parts comes from the product rule run backwards: ∫u dv = uv − ∫v du. Forgetting the minus sign in front of the second term is the single most common error.

4. Q4Easy

   In substitution, u = g(x) implies du = ?

   A.g'(x)

   B.g(x) dx

   C.g'(x) dx✓ Correct

   D.dx

   Answer: C. g'(x) dx

   Explanation: Differentiating u = g(x) gives du/dx = g'(x), so as a differential du = g'(x) dx — the dx must always travel with the derivative, not be left out.

5. Q5Easy

   ∫_a^b f(x) dx (definite integral) gives:

   A.A number (area / net signed area)✓ Correct

   B.A function of x

   C.Same as antiderivative

   D.Indefinite integral

   Answer: A. A number (area / net signed area)

   Explanation: A definite integral has fixed limits, so evaluating the antiderivative between them collapses to a single number (net signed area) — unlike an indefinite integral, still a function of x.

6. Q6Easy

   ∫ [f(x) + g(x)] dx =

   A.f + g

   B.∫ f · g dx

   C.∫ f − ∫ g

   D.∫ f dx + ∫ g dx✓ Correct

   Answer: D. ∫ f dx + ∫ g dx

   Explanation: Integration distributes over addition just like differentiation does: ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx. It does not distribute over a product, so don't try this with f(x)·g(x).

7. Q7Medium

   ∫ (1/x) dx = (for x > 0)

   A.x + C

   B.x²/2 + C

   C.1/x² + C

   D.ln |x| + C✓ Correct

   Answer: D. ln |x| + C

   Explanation: The power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) breaks down at n = −1 since that means dividing by 0 — this is the one case handled separately: ∫dx/x = ln|x| + C.

8. Q8Medium

   ∫ x² dx =

   A.x³/3 + C✓ Correct

   B.x²

   C.2x

   D.x²/2

   Answer: A. x³/3 + C

   Explanation: Power rule: raise the exponent from 2 to 3 and divide by the new exponent, giving x³/3 + C. Writing just x³ or x²/2 means forgetting to divide by the new exponent.

9. Q9Medium

   ∫ x cos x dx =

   A.cos x − x sin x

   B.x sin x

   C.x sin x + cos x + C✓ Correct

   D.sin x

   Answer: C. x sin x + cos x + C

   Explanation: Integration by parts with u = x, dv = cos x dx gives uv − ∫v du = x sin x − ∫sin x dx = x sin x + cos x + C. Forgetting the second term, ∫sin x dx, is the usual gap.

10. Q10Medium

    ∫ 2x · e^(x²) dx =

    A.e^(x²) + C✓ Correct

    B.e^(x²)/2

    C.x² · e^(x²)

    D.Cannot compute

    Answer: A. e^(x²) + C

    Explanation: Substitute u = x², so du = 2x dx matches the 2x factor already present: ∫eᵘ du = eᵘ + C = e^(x²) + C. Without that 2x, this substitution wouldn't cleanly work.

11. Q11Medium

    ∫_0^1 x² dx =

    A.1

    B.1/3✓ Correct

    C.1/2

    D.2/3

    Answer: B. 1/3

    Explanation: Antiderivative is x³/3; evaluate at the limits: 1³/3 − 0³/3 = 1/3. Writing just x³ without dividing by 3 is what produces the wrong answer of 1.

12. Q12Hard

    ∫ x² eˣ dx =

    A.x eˣ − eˣ

    B.x² eˣ

    C.eˣ(x² + 2)

    D.x² eˣ − 2x eˣ + 2 eˣ + C✓ Correct

    Answer: D. x² eˣ − 2x eˣ + 2 eˣ + C

    Explanation: Integration by parts twice: first u = x², dv = eˣ dx gives x²eˣ − 2∫x eˣ dx, then a second pass on ∫x eˣ dx = x eˣ − eˣ gives the full result x²eˣ − 2x eˣ + 2eˣ + C.

13. Q13Hard

    ∫ x/(x² + 1) dx =

    A.ln x

    B.tan⁻¹ x

    C.x²/2

    D.(1/2) ln(x² + 1) + C✓ Correct

    Answer: D. (1/2) ln(x² + 1) + C

    Explanation: Substitute u = x²+1, du = 2x dx, so x dx = du/2: (1/2)∫du/u = (1/2)ln|u| + C = (1/2)ln(x²+1) + C. Forgetting the ÷2 from du = 2x dx gives twice the correct answer.

14. Q14Hard

    ∫_0^(π/4) tan x dx =

    A.1

    B.ln 2

    C.(1/2) ln 2✓ Correct

    D.π/4

    Answer: C. (1/2) ln 2

    Explanation: Antiderivative of tan x is −ln|cos x|; evaluating from 0 to π/4 gives −ln(1/√2) − (−ln 1) = ln√2 = (1/2)ln 2. Forgetting cos(π/4) = 1/√2 is the usual slip.

Integrals — FAQs

More Class 12 Mathematics chapters

• 01Relations and Functions
• 02Inverse Trigonometric Functions
• 03Matrices
• 04Determinants
• 05Continuity and Differentiability
• 06Application of Derivatives

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