JEE Mathematics Definite Integration MCQs Set 17

Practice MCQs for JEE Mathematics Definite Integration

Review structured MCQ sets for JEE Mathematics Definite Integration. Built according to official JEE guidelines, these downloadable questions support daily revision and core concept reinforcement.

Access Definite Integration Questions and Solutions

Navigate directly to the 50 objective questions for Definite Integration using the digital viewer below. Each practice set includes verified answer keys, allowing students to instantly cross-check their work and identify areas requiring further revision.

Multiple Choice Questions

Question 1: \( \int\limits_{0}^{\pi} \left|{\sin{\left(x \right)} - \cos{\left(x \right)}}\right|\, dx = \)
(a) \( 4 \sqrt{2} \)
(b) \( 2 \sqrt{2} \)
(c) \( 4 \sqrt{3} \)
(d) \( 3 \sqrt{2} \)
Show Answer & Explanation

Answer: (b) \( 2 \sqrt{2} \)

Explanation:
1. cos x - sin x changes sign at x = π/4.
2. ∫0^(π/4) (cos x - sin x) dx = √2 - 1 and ∫_(π/4)^π (sin x - cos x) dx = 1 + √2.
3. The sum is 2√2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 2: \( \int\limits_{0}^{100} \sin{\left(\pi \left(x - \left\lfloor{x}\right\rfloor\right) \right)}\, dx = \)
(a) \( \frac{100}{\pi} \)
(b) \( \frac{200}{\pi} \)
(c) \( 100 \pi \)
(d) \( 200 \pi \)
Show Answer & Explanation

Answer: (b) \( \frac{200}{\pi} \)

Explanation:
1. x - [x] is the fractional part, which has period 1.
2. ∫0^100 = 100∫0^1 sin πx dx = 100 · 2/π = 200/π.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 3: Evaluate \( \lim_{x\to0}\frac{\int_0^x\sin^3t\cos t\,dt}{x^4} \) =
(a) \( 0.25 \)
(b) \( 2.5 \)
(c) \( 5.2 \)
(d) \( 0.52 \)
Show Answer & Explanation

Answer: (a) \( 0.25 \)

Explanation:
1. ∫0^x sin³t cos t dt = sin⁴x/4.
2. The limit is lim sin⁴x/(4x⁴) = 1/4 = 0.25.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 4: \( \int\limits_{0}^{\log{\left(2 \right)}} \cosh{\left(2 x \right)}\, dx = \)
(a) \( \frac{15}{16} \)
(b) \( -15 \)
(c) \( \frac{16}{17} \)
(d) \( \frac{17}{18} \)
Show Answer & Explanation

Answer: (a) \( \frac{15}{16} \)

Explanation:
1. ∫cosh 2x dx = (1/2) sinh 2x.
2. sinh(2 log 2) = (4 - 1/4)/2 = 15/8, so the value is 15/16.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 5: \( \int\limits_{0}^{\frac{\pi}{2}} \frac{\sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}\, dx = \)
(a) \( \frac{\pi}{2} \)
(b) \( 2 - \frac{\pi}{2} \)
(c) \( -2 + \frac{\pi}{2} \)
(d) \( \frac{\pi}{2} + 2 \)
Show Answer & Explanation

Answer: (b) \( 2 - \frac{\pi}{2} \)

Explanation:
1. sin²x/(1 + cos x)² = tan²(x/2) = sec²(x/2) - 1.
2. ∫ = [2 tan(x/2) - x] from 0 to π/2 = 2 - π/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 6: \( \int\limits_{1}^{e} \frac{\log{\left(x \right)}}{x^{2}}\, dx = \)
(a) \( 1 - \frac{2}{e} \)
(b) \( -1 + \frac{2}{e} \)
(c) \( \frac{-1 + e}{e} \)
(d) \( \frac{2}{e} + 1 \)
Show Answer & Explanation

Answer: (a) \( 1 - \frac{2}{e} \)

Explanation:
1. By parts: ∫log x/x² dx = -log x/x - 1/x.
2. From 1 to e: (-1/e - 1/e) - (0 - 1) = 1 - 2/e.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 7: \( \int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \sin{\left(\log{\left(x + \sqrt{x^{2} + 1} \right)} \right)}\, dx = \)
(a) \( 1 \)
(b) \( -1 \)
(c) \( 0 \)
(d) \( 2 \)
Show Answer & Explanation

Answer: (c) \( 0 \)

Explanation:
1. log(x + √(x² + 1)) is odd, so the sine of it is odd.
2. Integral over a symmetric interval = 0.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 8: \( \int\limits_{0}^{\frac{\pi}{2}} \frac{1}{\tan^{3}{\left(x \right)} + 1}\, dx = \)
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{2} \)
(c) \( 1 \)
(d) \( \log{\left(2 \right)} \)
Show Answer & Explanation

Answer: (a) \( \frac{\pi}{4} \)

Explanation:
1. Replace x by π/2 - x: the integrand becomes tan³x/(1 + tan³x).
2. Adding, 2I = ∫0^(π/2) dx = π/2, so I = π/4.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 9: \( \int\limits_{e^{-1}}^{e^{2}} \frac{\left|{\log{\left(x \right)}}\right|}{x}\, dx = \)
(a) \( \frac{3}{2} \)
(b) \( \frac{5}{2} \)
(c) \( 3 \)
(d) \( 5 \)
Show Answer & Explanation

Answer: (b) \( \frac{5}{2} \)

Explanation:
1. Put log x = t, so dx/x = dt and the limits become -1 to 2.
2. ∫_(-1)^2 |t| dt = 1/2 + 2 = 5/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 10: \( \int\limits_{0}^{1} \log{\left(-1 + \frac{1}{x} \right)}\, dx = \)
(a) \( 0 \)
(b) \( - \frac{1}{2} \)
(c) \( \frac{1}{3} \)
(d) \( \frac{1}{2} \)
Show Answer & Explanation

Answer: (a) \( 0 \)

Explanation:
1. log(1/x - 1) = log(1 - x) - log x.
2. ∫0^1 log(1 - x) dx = ∫0^1 log x dx = -1, so the difference is 0.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 11: If \( f(t)=\int_{-t}^{t}\frac{e^{-|x|}}2dx \), then \( \lim_{t\to\infty}f(t) \) =
(a) \( 1 \)
(b) \( \frac12 \)
(c) \( 0 \)
(d) \( -1 \)
Show Answer & Explanation

Answer: (a) \( 1 \)

Explanation:
1. The integrand is even, so f(t) = ∫0^t e^(-x) dx = 1 - e^(-t).
2. As t → ∞, f(t) → 1.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 12: \( \int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)} + \cos^{3}{\left(x \right)}}\, dx = \)
(a) \( \frac{\pi}{3} \)
(b) \( - \frac{\pi}{2} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{\pi}{12} \)
Show Answer & Explanation

Answer: (d) \( \frac{\pi}{12} \)

Explanation:
1. Replace x by π/6 + π/3 - x = π/2 - x: the integrand becomes sin³x/(sin³x + cos³x).
2. Adding, 2I = π/3 - π/6 = π/6, so I = π/12.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 13: \( \int\limits_{-2}^{2} \left(x^{11} \cos{\left(x \right)} + e^{x}\right)\, dx = \)
(a) \( \sinh{\left(2 \right)} \)
(b) \( 2 \sinh{\left(2 \right)} \)
(c) \( \frac{3 \sinh{\left(2 \right)}}{2} \)
(d) \( \frac{\sinh{\left(2 \right)}}{2} \)
Show Answer & Explanation

Answer: (b) \( 2 \sinh{\left(2 \right)} \)

Explanation:
1. x¹¹ cos x is odd, so its integral over [-2, 2] is 0.
2. ∫_(-2)^2 e^x dx = e² - e^(-2) = 2 sinh 2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 14: \( \int\limits_{0}^{\frac{\pi^{2}}{4}} \cos{\left(\sqrt{x} \right)}\, dx = \)
(a) \( 2 \)
(b) \( -2 + \pi \)
(c) \( 2 + \pi \)
(d) \( -1 + \frac{\pi}{2} \)
Show Answer & Explanation

Answer: (b) \( -2 + \pi \)

Explanation:
1. Put √x = t, so dx = 2t dt.
2. 2∫0^(π/2) t cos t dt = 2[t sin t + cos t] = 2(π/2 - 1) = π - 2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 15: \( \int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \log{\left(\frac{2 - \sin{\left(x \right)}}{\sin{\left(x \right)} + 2} \right)}\, dx = \)
(a) \( 0 \)
(b) \( 1 \)
(c) \( 2 \)
(d) \( -1 \)
Show Answer & Explanation

Answer: (a) \( 0 \)

Explanation:
1. Let g(x) be the integrand; g(-x) = log((2 + sin x)/(2 - sin x)) = -g(x), so g is odd.
2. The integral over a symmetric interval is 0.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 16: Let \( I_1=\int_1^2x\left[\sqrt x+\sqrt{3-x}\right]dx \) and \( I_2=\int_1^2\left(\sqrt x+\sqrt{3-x}\right)dx \). Then \( \frac{I_1}{I_2} \) =
(a) \( \frac12 \)
(b) \( \frac32 \)
(c) \( 2 \)
(d) \( 1 \)
Show Answer & Explanation

Answer: (b) \( \frac32 \)

Explanation:
1. The bracket is symmetric under x → 3 - x (since 1 + 2 = 3).
2. So I₁ = ∫(3 - x)[...] dx = 3I₂ - I₁, giving I₁ = (3/2)I₂.
3. Hence I₁/I₂ = 3/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 17: \( \int\limits_{0}^{\frac{\pi}{2}} \sqrt{\sin{\left(x \right)} + 1}\, dx = \)
(a) \( 1 \)
(b) \( \frac{1}{2} \)
(c) \( 2 \)
(d) \( 3 \)
Show Answer & Explanation

Answer: (c) \( 2 \)

Explanation:
1. 1 + sin x = (sin(x/2) + cos(x/2))², so the integrand is sin(x/2) + cos(x/2).
2. ∫ = [-2cos(x/2) + 2 sin(x/2)] from 0 to π/2 = 2(−cos(π/4) + sin(π/4)) + 2 = 2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 18: If \( \int_0^{40}\frac{dx}{2x+1}=\log a \), then \( a \) =
(a) \( 3 \)
(b) \( 9 \)
(c) \( 81 \)
(d) \( 40 \)
Show Answer & Explanation

Answer: (b) \( 9 \)

Explanation:
1. ∫dx/(2x + 1) = (1/2) log(2x + 1).
2. From 0 to 40: (1/2) log 81 = log 81^(1/2) = log 9.
3. So a = 9.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 19: \( \int\limits_{0}^{\frac{\pi}{6}} \sin^{2}{\left(6 x \right)} \cos^{4}{\left(3 x \right)}\, dx = \)
(a) \( \frac{5}{192} \)
(b) \( \frac{5 \pi}{256} \)
(c) \( \frac{5 \pi}{192} \)
(d) \( \frac{\pi}{96} \)
Show Answer & Explanation

Answer: (c) \( \frac{5 \pi}{192} \)

Explanation:
1. Write sin²6x = 4 sin²3x cos²3x, so the integrand is 4cos⁶3x sin²3x.
2. Put 3x = t: (4/3)∫0^(π/2) cos⁶t sin²t dt = (4/3)(5·3·1·1)/(8·6·4·2)·π/2 = (4/3)(15/384)(π/2) = 5π/192.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 20: If \( f(x)=\int_{x^2}^{x^4}\sin\sqrt t\,dt \), then \( f'(x) \) equals
(a) \( \sin x^2-\sin x \)
(b) \( 4x^3\sin x^2-2x\sin x \)
(c) \( x^2\sin x^2-x\sin x \)
(d) \( \sin x-\cos x \)
Show Answer & Explanation

Answer: (b) \( 4x^3\sin x^2-2x\sin x \)

Explanation:
1. By the Leibnitz rule, f'(x) = sin√(x⁴)·(4x³) - sin√(x²)·(2x).
2. √(x⁴) = x² and √(x²) = x for x ≥ 0.
3. So f'(x) = 4x³ sin x² - 2x sin x.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 21: If \( 0(a) \( \log\frac bc-\log\frac ac \)
(b) \( \frac{\log a-\log b}{\log c} \)
(c) \( \frac1{\log b/c}-\frac1{\log a/c} \)
(d) \( \log\frac ac-\log\frac bc \)
Show Answer & Explanation

Answer: (c) \( \frac1{\log b/c}-\frac1{\log a/c} \)

Explanation:
1. The integrand is (a/c)^x - (b/c)^x, with 0 < a/c < 1 and 0 < b/c < 1.
2. ∫0^∞ k^x dx = -1/log k for 0 < k < 1.
3. The value is -1/log(a/c) + 1/log(b/c) = 1/log(b/c) - 1/log(a/c).

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 22: \( \int\limits_{1}^{2} x^{2 x} \left(\log{\left(x \right)} + 1\right)\, dx = \)
(a) \( \frac{9}{2} \)
(b) \( \frac{11}{2} \)
(c) \( \frac{13}{2} \)
(d) \( \frac{15}{2} \)
Show Answer & Explanation

Answer: (d) \( \frac{15}{2} \)

Explanation:
1. d/dx(x^(2x)) = x^(2x)(2 log x + 2) = 2x^(2x)(1 + log x).
2. So ∫ = (1/2)[x^(2x)]₁² = (1/2)(16 - 1) = 15/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 23: \( \int\limits_{2}^{e} \left(\frac{1}{\log{\left(x \right)}} - \frac{1}{\log{\left(x \right)}^{2}}\right)\, dx = \)
(a) \( -2 + e \)
(b) \( e + \frac{2}{\log{\left(2 \right)}} \)
(c) \( e - \frac{2}{\log{\left(2 \right)}} \)
(d) \( \log{\left(2 \right)} \)
Show Answer & Explanation

Answer: (c) \( e - \frac{2}{\log{\left(2 \right)}} \)

Explanation:
1. d/dx[x/log x] = 1/log x - 1/(log x)², so the integrand is this derivative.
2. [x/log x] from 2 to e = e - 2/log 2 = e - 2 log₂e.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 24: \( \int\limits_{0}^{1} \frac{4 x^{3}}{\sqrt{1 - x^{8}}}\, dx = \)
(a) \( \pi \)
(b) \( - \pi \)
(c) \( \frac{\pi}{2} \)
(d) \( - \frac{\pi}{2} \)
Show Answer & Explanation

Answer: (c) \( \frac{\pi}{2} \)

Explanation:
1. Put x⁴ = t, so 4x³ dx = dt.
2. ∫0^1 dt/√(1 - t²) = sin⁻¹1 = π/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Free study material for Integrals

Multiple Choice Questions (MCQs) for JEE Mathematics Definite Integration

Download Multiple Choice Questions: Definite Integration (JEE Mathematics)

Explore reliable practice questions for Definite Integration tailored for JEE Mathematics learners. Use these multiple-choice formats to evaluate preparedness and strengthen problem-solving skills.

Concept Clarification for Definite Integration

Cross-reference your completed choices with comprehensive NCERT solutions for JEE Mathematics to ensure absolute clarity across all sub-topics in this chapter.

Additional Study Resources for JEE Mathematics

Explore our broader library of printable assignments, chapter notes, and mock tests designed to support continuous revision and secure higher marks in JEE assessments.

FAQs

Where can I access latest JEE Mathematics Definite Integration MCQs Set 17?

You can get most exhaustive JEE Mathematics Definite Integration MCQs Set 17 for free on StudiesToday.com. These MCQs for JEE Mathematics are updated for the 2026-27 academic session as per JEE examination standards.

Are Assertion-Reasoning and Case-Study MCQs included in the Mathematics JEE material?

Yes, our JEE Mathematics Definite Integration MCQs Set 17 include the latest type of questions, such as Assertion-Reasoning and Case-based MCQs. 50% of the JEE paper is now competency-based.

How do practicing Mathematics MCQs help in scoring full marks in JEE exams?

By solving our JEE Mathematics Definite Integration MCQs Set 17, JEE students can improve their accuracy and speed which is important as objective questions provide a chance to secure 100% marks in the Mathematics.

Do you provide answers and explanations for JEE Mathematics Definite Integration MCQs Set 17?

Yes, Mathematics MCQs for JEE have answer key and brief explanations to help students understand logic behind the correct option as its important for 2026 competency-focused JEE exams.

Can I practice these Mathematics JEE MCQs online?

Yes, you can also access online interactive tests for JEE Mathematics Definite Integration MCQs Set 17 on StudiesToday.com as they provide instant answers and score to help you track your progress in Mathematics.