JEE Mathematics Definite Integration MCQs Set 28

Download JEE MCQs for JEE Mathematics: Definite Integration

Access targeted multiple-choice questions for Definite Integration designed to align with the latest JEE academic syllabus for JEE Mathematics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.

Chapter-wise Objective Questions: Definite Integration

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: \( \lim_{n\to\infty}\sum_{r=1}^{n}\frac{1}{\sqrt{nr}} \) is equal to
(a) 2
(b) 1
(c) 0
(d) none of these
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. Σ 1/√(nr) = (1/n)Σ 1/√(r/n).
2. The limit is ∫₀¹ x^(−1/2) dx = 2.

Teacher's Note:
1. Convert the sum into a definite integral with r/n → x and 1/n → dx.
2. Evaluate the integral.

Question 2: \( \lim_{n\to\infty}\sum_{r=0}^{n-1}\frac{1}{\sqrt{n^2-r^2}} \) is
(a) π
(b) π/2
(c) π/4
(d) none of these
Show Answer & Explanation

Answer: (b) π/2

Explanation:
1. Σ 1/√(n² − r²) = (1/n)Σ 1/√(1 − (r/n)²).
2. The limit is ∫₀¹ dx/√(1 − x²) = sin⁻¹1 = π/2.

Teacher's Note:
1. Convert the sum into a definite integral with r/n → x and 1/n → dx.
2. Evaluate the integral.

Question 3: \( \lim_{n\to\infty}\sum_{r=1}^{n}\frac1ne^{r/n} \) is
(a) e
(b) e − 1
(c) 1 − e
(d) e + 1
Show Answer & Explanation

Answer: (b) e − 1

Explanation:
1. The limit is ∫₀¹ eˣ dx = e − 1.

Teacher's Note:
1. Convert the sum into a definite integral with r/n → x and 1/n → dx.
2. Evaluate the integral.

Question 4: \( \lim_{n\to\infty}\sum_{r=1}^{n}\frac1n\sin\frac{r\pi}{2n} \) is
(a) π/2
(b) 2
(c) 2/π
(d) none of these
Show Answer & Explanation

Answer: (c) 2/π

Explanation:
1. The limit is ∫₀¹ sin(πx/2) dx = [−(2/π)cos(πx/2)]₀¹ = 2/π.

Teacher's Note:
1. Convert the sum into a definite integral with r/n → x and 1/n → dx.
2. Evaluate the integral.

Question 5: \( \lim_{n\to\infty}\sum_{r=1}^{4n}\frac{1}{n+r} \) is
(a) logₑ 5
(b) 0
(c) logₑ 4
(d) none of these
Show Answer & Explanation

Answer: (a) logₑ 5

Explanation:
1. Σ 1/(n + r) = (1/n)Σ 1/(1 + r/n) with r up to 4n, so x runs from 0 to 4.
2. The limit is ∫₀⁴ dx/(1 + x) = ln 5.

Teacher's Note:
1. Convert the sum into a definite integral with r/n → x and 1/n → dx.
2. Evaluate the integral.

Question 6: \( \lim_{n\to\infty}\frac1n\sum_{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}} \) equals
(a) \( 1+\sqrt5 \)
(b) \( -1+\sqrt5 \)
(c) \( -1+\sqrt2 \)
(d) \( 1+\sqrt2 \)
Show Answer & Explanation

Answer: (b) \( -1+\sqrt5 \)

Explanation:
1. The limit is ∫₀² x/√(1 + x²) dx = [√(1 + x²)]₀² = √5 − 1.

Teacher's Note:
1. Convert the sum into a definite integral with r/n → x and 1/n → dx.
2. Evaluate the integral.

Question 7: \( \lim_{n\to\infty}\frac{2^k+4^k+6^k+\cdots+(2n)^k}{n^{k+1}} \), k ≠ −1, is equal to
(a) 2ᵏ
(b) \( \frac{2^k}{k+1} \)
(c) \( \frac{1}{k+1} \)
(d) none of these
Show Answer & Explanation

Answer: (b) \( \frac{2^k}{k+1} \)

Explanation:
1. The sum is 2ᵏ Σ rᵏ, so the expression is 2ᵏ(1/n)Σ (r/n)ᵏ.
2. The limit is 2ᵏ∫₀¹ xᵏ dx = 2ᵏ/(k + 1).

Teacher's Note:
1. Convert the sum into a definite integral with r/n → x and 1/n → dx.
2. Evaluate the integral.

Question 8: If \( g(x)=\int_0^x\cos^4t\,dt \), then g(x + π) equals
(a) g(x) + g(π)
(b) g(x) − g(π)
(c) g(x)g(π)
(d) g(x)/g(π)
Show Answer & Explanation

Answer: (a) g(x) + g(π)

Explanation:
1. g(x + π) = g(π) + ∫_π^(x+π) cos⁴t dt.
2. cos⁴t has period π, so the last integral equals ∫₀ˣ cos⁴t dt = g(x).

Teacher's Note:
1. Use the periodicity of cos⁴t.

Question 9: \( \int_0^2\sqrt{\frac{2+x}{2-x}}\,dx \) is equal to
(a) π + 1
(b) 1 + π/2
(c) π + 3/2
(d) none of these
Show Answer & Explanation

Answer: (d) none of these

Explanation:
1. Write √((2 + x)/(2 − x)) = (2 + x)/√(4 − x²).
2. ∫₀² 2/√(4 − x²) dx = 2·(π/2) = π and ∫₀² x/√(4 − x²) dx = 2.
3. The value is π + 2, which is not among the first three options.

Teacher's Note:
1. Multiply numerator and denominator by √(2 + x).

Question 10: \( \int_1^{e}\frac{\pi\sin(\pi\log_ex)}{x}dx \) is equal to
(a) 2
(b) −2
(c) 2/π
(d) 2π
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. Put u = π ln x, du = (π/x) dx; the limits become 0 to π.
2. ∫₀^π sin u du = 2.

Teacher's Note:
1. Substitute for the logarithm.

Question 11: The value of \( \int_1^2[f\{g(x)\}]^{-1}f'\{g(x)\}g'(x)\,dx \), where g(1) = g(2), is equal to
(a) 1
(b) 2
(c) 0
(d) none of these
Show Answer & Explanation

Answer: (c) 0

Explanation:
1. The integrand is d/dx ln f(g(x)).
2. The integral = ln f(g(2)) − ln f(g(1)) = 0 because g(1) = g(2).

Teacher's Note:
1. Recognise the logarithmic derivative of f(g(x)).

Question 12: \( \int_{\pi/4}^{3\pi/4}\frac{dx}{1+\cos x} \) is equal to
(a) 2
(b) −2
(c) 1/2
(d) −1/2
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. 1 + cos x = 2cos²(x/2), so the integral is ∫½sec²(x/2)dx = tan(x/2).
2. tan(3π/8) − tan(π/8) = (√2 + 1) − (√2 − 1) = 2.

Teacher's Note:
1. Use 1 + cos x = 2cos²(x/2).

Question 13: If f(x) satisfies the conditions of Rolle's theorem in [1, 2], then \( \int_1^2f'(x)\,dx \) is equal to
(a) 1
(b) 3
(c) 0
(d) none of these
Show Answer & Explanation

Answer: (c) 0

Explanation:
1. ∫₁² f′(x) dx = f(2) − f(1).
2. Rolle's theorem requires f(1) = f(2), so the integral is 0.

Teacher's Note:
1. Rolle's theorem gives equal end values.

Free study material for Integrals

Practice MCQs for JEE Mathematics Definite Integration

Download Multiple Choice Questions: Definite Integration (JEE Mathematics)

Test your conceptual understanding of Definite Integration with these targeted multiple-choice questions. Designed in alignment with the latest JEE curriculum for JEE Mathematics, these problem sets build accuracy and prepare students for objective exams.

Concept Clarification for Definite Integration

Each question includes structured solution keys mapped directly to standard JEE textbooks, helping students evaluate their reasoning and correct mistakes early in their revision.

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