JEE Mathematics Definite Integration MCQs Set 16

Mathematics Objective Questions and Answers: 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.

Download Definite Integration MCQs with Answers

Access the complete set of multiple-choice questions for Definite Integration below. This focused format allows students to isolate specific topics for thorough review and uninterrupted practice alongside official JEE textbooks.

Multiple Choice Questions

Question 1: \( \int\limits_{0}^{1} \frac{1}{\sqrt{4 - x^{2}}}\, dx = \)
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{\pi}{5} \)
Show Answer & Explanation

Answer: (c) \( \frac{\pi}{6} \)

Explanation:
1. The sum Σ 1/√(4n² - r²) = (1/n)Σ 1/√(4 - (r/n)²) tends to ∫0^1 dx/√(4 - x²).
2. = [sin⁻¹(x/2)] from 0 to 1 = π/6.

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

Answer: (d) \( 1 + \frac{\pi}{2} \)

Explanation:
1. The limit is ∫0^1 √((1 + x)/(1 - x)) dx = ∫0^1 (1 + x)/√(1 - x²) dx.
2. = [sin⁻¹x - √(1 - x²)] from 0 to 1 = π/2 + 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 3: \( \int\limits_{0}^{\frac{\pi}{2}} \sin{\left(2 x \right)} \cos^{5}{\left(x \right)}\, dx = \)
(a) \( \frac{2}{7} \)
(b) \( \frac{1}{7} \)
(c) \( - \frac{1}{7} \)
(d) \( \frac{3}{7} \)
Show Answer & Explanation

Answer: (a) \( \frac{2}{7} \)

Explanation:
1. sin 2x = 2 sin x cos x, so the integrand is 2cos⁶x sin x.
2. ∫0^(π/2) 2cos⁶x sin x dx = 2[-cos⁷x/7] = 2/7.

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: If \( \int_n^{n+1}f(x)dx=n^2+n \), then \( \int_{-1}^{1}f(x)dx \) =
(a) \( 0 \)
(b) \( -2 \)
(c) \( 3 \)
(d) \( 4 \)
Show Answer & Explanation

Answer: (a) \( 0 \)

Explanation:
1. Put n = -1: ∫_(-1)^0 f(x) dx = 1 - 1 = 0.
2. Put n = 0: ∫0^1 f(x) dx = 0.
3. So ∫_(-1)^1 f(x) dx = 0 + 0 = 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 5: \( \int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{4}{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = \)
(a) \( \frac{3 \pi}{128} \)
(b) \( \frac{3 \pi}{256} \)
(c) \( \frac{5 \pi}{128} \)
(d) \( \frac{7 \pi}{128} \)
Show Answer & Explanation

Answer: (b) \( \frac{3 \pi}{256} \)

Explanation:
1. The integrand is even, so the integral is 2∫0^(π/2) sin⁴x cos⁶x dx.
2. ∫0^(π/2) sin⁴x cos⁶x dx = 3π/512.
3. Value = 3π/256.

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: If \( f(x)=\int_{-1}^{x}|t|\,dt \), then for any \( x\ge0 \), \( f(x) \) =
(a) \( \frac12(1-x^2) \)
(b) \( 1-x^2 \)
(c) \( \frac12(1+x^2) \)
(d) \( 1+x^2 \)
Show Answer & Explanation

Answer: (c) \( \frac12(1+x^2) \)

Explanation:
1. For x ≥ 0: ∫_(-1)^0 (-t) dt + ∫0^x t dt = 1/2 + x²/2.
2. So f(x) = (1/2)(1 + x²).

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: If \( I_1=\int_0^{3\pi}f(\cos^2x)dx \) and \( I_2=\int_0^{\pi}f(\cos^2x)dx \), then
(a) \( I_1=I_2 \)
(b) \( I_1=2I_2 \)
(c) \( I_1=5I_2 \)
(d) \( I_1=3I_2 \)
Show Answer & Explanation

Answer: (d) \( I_1=3I_2 \)

Explanation:
1. f(cos²x) has period π.
2. The integral over a length of 3π is three times the integral over one period.
3. So I₁ = 3I₂.

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: Evaluate \( \int_0^a\frac{x^5}{\sqrt{a^2-x^2}}dx \) =
(a) \( \frac{a^5}{15} \)
(b) \( \frac{8a^5}{15} \)
(c) \( \frac{8a}{15} \)
(d) \( \frac{11a^2}{15} \)
Show Answer & Explanation

Answer: (b) \( \frac{8a^5}{15} \)

Explanation:
1. Put x = a sin θ: the integral is a⁵∫0^(π/2) sin⁵θ dθ.
2. ∫0^(π/2) sin⁵θ dθ = (4·2)/(5·3) = 8/15.
3. Value = 8a⁵/15.

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

Answer: (a) \( -1 + 2 \log{\left(2 \right)} \)

Explanation:
1. ∫log x dx = x log x - x.
2. [x log x - x] from 1 to 2 = (2 log 2 - 2) - (-1) = 2 log 2 - 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 10: \( \int\limits_{\frac{1}{\pi}}^{\frac{2}{\pi}} \frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}}\, dx = \)
(a) \( \frac{\pi}{2} \)
(b) \( \pi \)
(c) \( 1 \)
(d) \( 0 \)
Show Answer & Explanation

Answer: (c) \( 1 \)

Explanation:
1. Put 1/x = t, so dx/x² = -dt.
2. The limits become π to π/2, so the integral is ∫_(π/2)^π sin t dt = cos(π/2) - cos π = 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 11: The function \( F(x)=\int_0^x\log\left(t+\sqrt{1+t^2}\right)dt \) is
(a) \( a periodic function \)
(b) \( an even function \)
(c) \( an odd function \)
(d) \( neither even nor odd \)
Show Answer & Explanation

Answer: (b) \( an even function \)

Explanation:
1. The integrand g(t) = log(t + √(1 + t²)) is odd.
2. The integral of an odd function from 0 to x is an even function of x.
3. So F(-x) = F(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 12: Evaluate \( \frac1\pi\int_{-2}^{2}\frac{dx}{4+x^2} \) =
(a) \( 0 \)
(b) \( \frac14 \)
(c) \( \frac\pi4 \)
(d) \( \frac12 \)
Show Answer & Explanation

Answer: (b) \( \frac14 \)

Explanation:
1. ∫dx/(4 + x²) = (1/2) tan⁻¹(x/2).
2. (1/π)(1/2)[tan⁻¹1 - tan⁻¹(-1)] = (1/π)(1/2)(π/2) = 1/4.

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 13: If \( I=\int_{-a}^{a}\left(\alpha\sin^5x+\beta\tan^3x+\gamma\cos x\right)dx \), where \( \alpha,\beta,\gamma \) are constants, then the value of \( I \) depends on
(a) \( \gamma,\ a \)
(b) \( \alpha,\beta,\gamma,\ a \)
(c) \( \alpha,\beta,\ a \)
(d) \( \beta \)
Show Answer & Explanation

Answer: (a) \( \gamma,\ a \)

Explanation:
1. sin⁵x and tan³x are odd functions, so their integrals over [-a, a] vanish.
2. cos x is even, so I = 2γ∫0^a cos x dx = 2γ sin a.
3. I depends on γ and a only.

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: If \( \int_0^k\frac{\cos x}{1+\sin^2x}dx=\frac\pi4 \), then \( k \) =
(a) \( 1 \)
(b) \( \frac\pi4 \)
(c) \( \frac\pi2 \)
(d) \( \frac\pi6 \)
Show Answer & Explanation

Answer: (c) \( \frac\pi2 \)

Explanation:
1. Put sin x = t: ∫cos x dx/(1 + sin²x) = tan⁻¹(sin x).
2. tan⁻¹(sin k) = π/4 gives sin k = 1.
3. The principal value is k = 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: If \( \int_{\sqrt2}^{x}\frac{dx}{x\sqrt{x^2-1}}=\frac\pi{12} \), then \( x \) =
(a) \( 1 \)
(b) \( \frac12 \)
(c) \( 2 \)
(d) \( -2 \)
Show Answer & Explanation

Answer: (c) \( 2 \)

Explanation:
1. ∫dx/(x√(x² - 1)) = sec⁻¹x.
2. sec⁻¹x - sec⁻¹√2 = π/12, so sec⁻¹x = π/4 + π/12 = π/3.
3. x = sec(π/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 16: Evaluate \( \lim_{n\to\infty}\left\{\frac1n+\frac1{n+1}+\frac1{n+2}+\dots+\frac1{3n}\right\} \) =
(a) \( \frac\pi4 \)
(b) \( \frac\pi6 \)
(c) \( \log2 \)
(d) \( \log3 \)
Show Answer & Explanation

Answer: (d) \( \log3 \)

Explanation:
1. The sum is Σ from r = 0 to 2n of 1/(n + r) = (1/n)Σ 1/(1 + r/n).
2. This tends to ∫0^2 dx/(1 + x) = log 3.

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: \( \int\limits_{0}^{1} x e^{x}\, dx = \)
(a) \( e \)
(b) \( 1 \)
(c) \( 2 e \)
(d) \( 2 \)
Show Answer & Explanation

Answer: (b) \( 1 \)

Explanation:
1. (1/n²)Σ r e^(r/n) = (1/n)Σ (r/n) e^(r/n) tends to ∫0^1 x e^x dx.
2. = [(x - 1)e^x] from 0 to 1 = 0 + 1 = 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 18: Evaluate \( \int_0^{1000}e^{x-[x]}dx \), where \( [\ ] \) denotes the greatest integer function.
(a) \( \frac{e^{1000}-1}{1000} \)
(b) \( \frac{e^{1000}-1}{e-1} \)
(c) \( 1000(e-1) \)
(d) \( \frac{e-1}{1000} \)
Show Answer & Explanation

Answer: (c) \( 1000(e-1) \)

Explanation:
1. x - [x] is periodic with period 1, so ∫0^1000 = 1000∫0^1 e^x dx.
2. The value is 1000(e - 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 19: The value of \( \int_{\alpha}^{\pi/2-\alpha}\frac{\cot x}{\tan x+\cot x}dx \) is
(a) \( \frac\pi4-\alpha \)
(b) \( \frac\pi2 \)
(c) \( \frac\pi3 \)
(d) \( \frac\pi8 \)
Show Answer & Explanation

Answer: (a) \( \frac\pi4-\alpha \)

Explanation:
1. Replace x by (π/2 - α + α) - x = π/2 - x: the integrand becomes tan x/(tan x + cot x).
2. Adding, 2I = (π/2 - α) - α = π/2 - 2α.
3. So I = π/4 - α.

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

Answer: (c) \( - \frac{\pi}{4} + \tan^{-1}{\left(e \right)} \)

Explanation:
1. Multiply by e^x: ∫e^x dx/(e^(2x) + 1) = tan⁻¹(e^x).
2. tan⁻¹e - tan⁻¹1 = tan⁻¹e - 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 21: If \( \int_{-1}^{4}f(x)dx=4 \) and \( \int_2^4(3-f(x))dx=7 \), then \( \int_{-1}^{2}f(x)dx \) =
(a) \( -2 \)
(b) \( 3 \)
(c) \( 5 \)
(d) \( 8 \)
Show Answer & Explanation

Answer: (c) \( 5 \)

Explanation:
1. ∫2^4 (3 - f) dx = 6 - ∫2^4 f dx = 7, so ∫2^4 f dx = -1.
2. ∫_(-1)^2 f dx = ∫_(-1)^4 f dx - ∫2^4 f dx = 4 - (-1) = 5.

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

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

Explanation:
1. By parts: ∫x e^(-x) dx = -x e^(-x) - e^(-x), and ∫e^(-x) dx = -e^(-x).
2. So ∫(x - 1)e^(-x) dx = -x e^(-x) - e^(-x) + e^(-x) = -x e^(-x).
3. [-x e^(-x)] from 0 to 1 = -1/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 23: \( \int\limits_{0}^{2} \left(\left|{x}\right| + \left|{x - 1}\right|\right)\, dx = \)
(a) \( 1 \)
(b) \( -1 \)
(c) \( 2 \)
(d) \( 3 \)
Show Answer & Explanation

Answer: (d) \( 3 \)

Explanation:
1. ∫0^2 |x| dx = 2.
2. ∫0^2 |x - 1| dx = 1/2 + 1/2 = 1.
3. The sum is 3.

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: \( \int\limits_{1}^{32} \frac{1}{\sqrt[5]{x} \sqrt{x^{\frac{4}{5}} + 1}}\, dx = \)
(a) \( \frac{2 \sqrt{2}}{5} + \frac{2 \sqrt{17}}{5} \)
(b) \( - \frac{2 \sqrt{2}}{5} + \frac{2 \sqrt{17}}{5} \)
(c) \( - \frac{5 \sqrt{2}}{2} + \frac{5 \sqrt{17}}{2} \)
(d) \( \frac{5 \sqrt{2}}{2} + \frac{5 \sqrt{17}}{2} \)
Show Answer & Explanation

Answer: (c) \( - \frac{5 \sqrt{2}}{2} + \frac{5 \sqrt{17}}{2} \)

Explanation:
1. Put x^(4/5) = t, so (4/5)x^(-1/5) dx = dt.
2. The integral is (5/4)∫dt/√(1 + t) = (5/2)√(1 + t) with t from 1 to 16.
3. Value = (5/2)(√17 - √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 25: If \( I_n=\int_0^\infty e^{-x}x^{n-1}dx \), then \( \int_0^\infty e^{-\lambda x}x^{n-1}dx \) is equal to
(a) \( \lambda I_n \)
(b) \( \frac1\lambda I_n \)
(c) \( \frac{I_n}{\lambda^n} \)
(d) \( \lambda^nI_n \)
Show Answer & Explanation

Answer: (c) \( \frac{I_n}{\lambda^n} \)

Explanation:
1. Put λx = t, so dx = dt/λ and x^(n-1) = t^(n-1)/λ^(n-1).
2. The integral becomes (1/λⁿ)∫0^∞ e^(-t) t^(n-1) dt.
3. Value = I_n/λⁿ.

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.

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