JEE Mathematics Definite Integration MCQs Set 26

Answer: (b) \( \log{\left(\frac{2 + \sqrt{5}}{1 + \sqrt{2}} \right)} \)

Explanation:
1. ∫dx/√(1 + x²) = log(x + √(1 + x²)).
2. From 1 to 2: log(2 + √5) - log(1 + √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.

Answer: (d) \( 0 \)

Explanation:
1. f(x) is even and sin nx is odd, so f(x) sin nx is odd.
2. The integral over [-π, π] 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.

Answer: (c) \( - \frac{3}{4} + \frac{e^{2}}{4} \)

Explanation:
1. e^x sinh x = (e^(2x) - 1)/2.
2. ∫0^1 = [e^(2x)/4 - x/2] = (e² - 1)/4 - 1/2 = (e² - 3)/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.

Answer: (a) \( [a]f(a)-\{f(1)+f(2)+\cdots+f([a])\} \)

Explanation:
1. Split at the integers 2, 3, ..., [a]: on [k, k + 1), [x] = k.
2. The sum is Σ k[f(k + 1) - f(k)] for k = 1 to [a] - 1, plus [a][f(a) - f([a])].
3. Collecting terms: [a]f(a) - {f(1) + f(2) + ... + f([a])}.

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.

Answer: (a) \( - \frac{3}{5} + \frac{3 e^{5}}{5} \)

Explanation:
1. {x} has period 1 and the interval has length 3.
2. I = 3∫0^1 e^(5x) dx = 3(e⁵ - 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.

Answer: (c) \( 200\sqrt2 \)

Explanation:
1. 1 + cos 2x = 2cos²x, so the integrand is √2 |cos x|, which has period π.
2. ∫0^(100π) = 100 ∫0^π √2 |cos x| dx = 100√2 (1 + 1).
3. The value is 200√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.

Answer: (d) \( - \frac{1}{3 e} + \frac{e^{2}}{3} \)

Explanation:
1. ∫e^(2 - 3x) dx = -(1/3)e^(2 - 3x).
2. From 0 to 1: -(1/3)(e^(-1) - e²) = (1/3)(e² - 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.

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

Explanation:
1. Each term is (1/n)√(1 - (r/n)²).
2. The limit is ∫0^1 √(1 - x²) dx, the area of a quarter of the unit circle = 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.

Answer: (d) \( - \log{\left(2 \right)} + \frac{\pi}{2} \)

Explanation:
1. Put x = tan θ: sin⁻¹(2x/(1 + x²)) = 2θ, so I = ∫0^(π/4) 2θ sec²θ dθ.
2. By parts: [2θ tan θ] - 2∫tan θ dθ = π/2 - 2·(1/2) log 2 = π/2 - log 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.

Answer: (a) \( \frac32 \)

Explanation:
1. The numerator is [(7 - x)²] and the denominator is [(7 - x)²] + [x²], which is positive on [2, 5].
2. Replace x by 2 + 5 - x = 7 - x: the integrand becomes [x²]/([x²] + [(7 - x)²]).
3. The two forms add to 1, so 2I = 3 and I = 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.

Answer: (a) \( \frac\pi4+\frac12\log2 \)

Explanation:
1. The sum equals (1/n)Σ (1 + r/n)/(1 + (r/n)²) for r = 1 to n.
2. In the limit it is ∫0^1 (1 + x)/(1 + x²) dx.
3. = [tan⁻¹x + (1/2) log(1 + x²)] from 0 to 1 = π/4 + (1/2) log 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.

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

Explanation:
1. Put x = tan θ; the limits are tan 15° and tan 75°, i.e. π/12 and 5π/12.
2. The integrand becomes tan θ/(1 + tan θ).
3. By θ → π/2 - θ, the pair adds to 1, so I = (1/2)(5π/12 - π/12) = 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.

Answer: (d) \( 0 \)

Explanation:
1. Put x = -t in the first integral: sin²⁷x is odd, so ∫_(-2n-1)^(-2n) sin²⁷x dx = -∫_(2n)^(2n+1) sin²⁷t dt.
2. Each term of the first sum cancels the corresponding term of the second.
3. The total 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.

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

Explanation:
1. Replace x by π/2 - x: sec³x ↔ cosec³x, so the two forms add to 1.
2. 2I = 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.

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

Explanation:
1. sec x/(sec x + cosec x) = sin x/(sin x + cos x).
2. By the property x → π/2 - x, the two forms add to 1.
3. 2I = 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.

Answer: (b) \( - \sqrt{3} - \sqrt{2} + 5 \)

Explanation:
1. [x²] = 0 on [0, 1), 1 on [1, √2), 2 on [√2, √3), 3 on [√3, 2).
2. ∫ = 0 + (√2 - 1) + 2(√3 - √2) + 3(2 - √3).
3. = 5 - √2 - √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.

Answer: (c) \( \frac\pi{60} \)

Explanation:
1. Write 1/((x² + 4)(x² + 9)) = x²/((x² + 0²)(x² + 2²)(x² + 3²)), so a = 0, b = 2, c = 3.
2. The value is π/(2(0 + 2)(2 + 3)(3 + 0)) = π/60.

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.

Answer: (a) \( \frac{34}{3} \)

Explanation:
1. x ± 2√(x - 1) = (√(x - 1) ± 1)² + ... so the roots are |√(x - 1) + 1| and |√(x - 1) - 1|.
2. On [1, 2] the sum is 2; on [2, 5] the sum is 2√(x - 1).
3. ∫1^2 2 dx + ∫2^5 2√(x - 1) dx = 2 + (4/3)(8 - 1) = 34/3.

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.

Answer: (d) \( 3025 \)

Explanation:
1. On [k, k + 1), [x]³ = k³.
2. ∫0^11 = 1³ + 2³ + ... + 10³ = (10·11/2)² = 3025.

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.

Answer: (d) \( \frac{5}{3} - \log{\left(4 \right)} \)

Explanation:
1. Put √x = t: ∫0^1 2t³/(1 + t) dt = 2∫0^1 (t² - t + 1 - 1/(1 + t)) dt.
2. = 2(1/3 - 1/2 + 1 - log 2) = 5/3 - 2 log 2 = 5/3 - log 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.

Answer: (c) \( \frac{4}{15} \)

Explanation:
1. sin³x cos²x is odd, so it integrates to 0.
2. sin²x cos³x is even, so I = 2∫0^(π/2) sin²x cos³x dx.
3. By Wallis, ∫0^(π/2) sin²x cos³x dx = (1·2)/(5·3·1) = 2/15, so I = 4/15.

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.

Answer: (b) \( 0 \)

Explanation:
1. 2^x + 2^(-x) is even and 3^x - 3^(-x) is odd.
2. The product is odd, so the integral over [-4, 4] 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.

Answer: (b) \( 102 - \tan{\left(1 \right)} \)

Explanation:
1. [tan⁻¹x] = 0 for 0 ≤ x < tan 1 and 1 for tan 1 ≤ x ≤ 102.
2. ∫0^102 = 102 - tan 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.