JEE Mathematics Definite Integration MCQs Set 25

Answer: (b) \( an A.P. \)

Explanation:
1. I_n + I_(n+2) = 1/(n + 1).
2. So the terms are 3, 4, 5, ...
3. They form an A.P. with common difference 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.

Answer: (b) \( \frac{23}{12} - \frac{2 \sqrt{2}}{3} \)

Explanation:
1. [x²] = 0 on [0, 1), 1 on [1, √2), 2 on [√2, 1.5].
2. ∫ = ∫1^√2 x² dx + 2∫_√2^1.5 x² dx = (2√2 - 1)/3 + (2/3)(3.375 - 2√2).
3. = (5.75 - 2√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) \( \frac{244140625}{132} \)

Explanation:
1. Replace x by 5 - x: ∫0^5 (5 - x)x^10 dx = [5x^11/11 - x^12/12] at 5.
2. = 5^12(1/11 - 1/12) = 5^12/132.

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) \( I₁ < I₂ \)

Explanation:
1. On (0, π/4) we have sin x < cos x.
2. Since 2^t is increasing, 2^(sin x) < 2^(cos x) throughout.
3. So I₁ < I₂.

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) \( 36 \)

Explanation:
1. ∫0^64 x^(1/3) dx = (3/4)(256) = 192.
2. ∫0^64 [∛x] dx = Σ_(k=0)^3 k((k + 1)³ - k³) = 0 + 7 + 38 + 111 = 156.
3. Value = 192 - 156 = 36.

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) \( \pi \log{\left(2 \right)} \)

Explanation:
1. Put x = tan θ: ∫0^(π/4) log(1 + tan θ) dθ = (π/8) log 2.
2. So 8 × (π/8) log 2 = π 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: (b) \( \frac{\pi}{6} \)

Explanation:
1. tan x - 1 changes sign at π/4.
2. ∫0^(π/4)(1 - tan x) dx = π/4 - (1/2) log 2.
3. ∫_(π/4)^(π/3)(tan x - 1) dx = (1/2) log 2 - π/12.
4. Total = 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: (b) \( \pi \)

Explanation:
1. Replace x by π - x and add: 2I = π∫0^π dx/(1 + sin x).
2. ∫0^π dx/(1 + sin x) = [tan x - sec x] type evaluation gives 2.
3. So 2I = 2π and I = 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.

Answer: (a) \( I>\frac12 \)

Explanation:
1. For 0 < x < 1/2 and n ≥ 1, x^(2n) ≤ x², so 1/√(1 - x^(2n)) > 1.
2. Hence I > ∫0^(1/2) 1 dx = 1/2.
3. (Also I < ∫0^(1/2) dx/√(1 - x²) = π/6, so I < 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.

Answer: (b) \( \frac{\sqrt{\pi} \sqrt{\frac{1}{a}}}{2} \)

Explanation:
1. Put t = √a x, so dx = dt/√a.
2. The integral is (1/√a)∫0^∞ e^(-t²) dt = (1/√a)(√π/2) = (1/2)√(π/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: (d) \( \frac1{10} \)

Explanation:
1. The expression is (1/n)Σ (r/n)^9.
2. The limit is ∫0^1 x^9 dx = 1/10.

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) \( \frac{\pi^2}4 \)

Explanation:
1. By L'Hospital's rule the limit is (tan⁻¹x)²/(x/√(x² + 1)).
2. As x → ∞, tan⁻¹x → π/2 and x/√(x² + 1) → 1.
3. The limit is π²/4.

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) \( \sqrt{\frac{15}8} \)

Explanation:
1. (∫√(1 + x)√(1 + x³) dx)² ≤ ∫(1 + x) dx · ∫(1 + x³) dx.
2. = (3/2)(5/4) = 15/8.
3. So the integral cannot exceed √(15/8). (The other values are smaller than the actual value, about 1.368.)

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) \( \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{2} \)

Explanation:
1. Replace x by π/2 - x and add: 2I = ∫0^(π/2) (sin²x + cos²x)/(sin x + cos x) dx = ∫0^(π/2) dx/(sin x + cos x).
2. sin x + cos x = √2 sin(x + π/4), so ∫ = (1/√2) ∫ cosec(x + π/4) dx = (1/√2)·2 log(√2 + 1)... giving 2I = √2 log(√2 + 1).
3. I = (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.

Answer: (d) \( \frac{8}{693} \)

Explanation:
1. Wallis formula with m = 6, n = 5: (5·3·1)(4·2)/(11·9·7·5·3·1) = 8/693.

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) \( for all values of a and b \)

Explanation:
1. Put t = a + b - x; as x goes from a to b, t goes from b to a, and dx = -dt.
2. The integral becomes ∫b^a f(t)(-dt) = ∫a^b f(t) dt.
3. This holds for all a and b.

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) \( I₁ = I₂ \)

Explanation:
1. Put t = sin θ + cos²θ: dt = (cos θ - 2 sin θ cos θ) dθ = (cos θ - sin 2θ) dθ.
2. I₁ - I₂ = ∫ f(t) dt with t = 1 at both θ = 0 and θ = π/2.
3. So I₁ - I₂ = 0 and I₁ = I₂.

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) \( \log4 \)

Explanation:
1. The r-th term is 1/(r + √(rn)) = (1/n)·1/(r/n + √(r/n)).
2. The limit is ∫0^1 dx/(x + √x).
3. Put √x = t: ∫0^1 2t dt/(t² + t) = 2∫0^1 dt/(t + 1) = 2 log 2 = 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: (d) \( 400 \)

Explanation:
1. (1 - cos 2x)/2 = sin²x, so the integrand is |sin x|, with period π.
2. The interval has length 200π, which is 200 periods.
3. ∫ = 200 ∫0^π sin x dx = 200 × 2 = 400.

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) \( less than 1 \)

Explanation:
1. On [2, 3], x³ ≥ 8, so 1 + x³ ≥ 9 and 1/√(1 + x³) ≤ 1/3.
2. So the integral over an interval of length 1 is at most 1/3, which is less than 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.

Answer: (d) \( 0 \)

Explanation:
1. (2x - 1)/(1 + x - x²) = (x - (1 - x))/(1 + x(1 - x)), so the integrand is tan⁻¹x - tan⁻¹(1 - x).
2. ∫0^1 tan⁻¹(1 - x) dx = ∫0^1 tan⁻¹x dx, so the two parts cancel and I = 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) \( \log{\left(6 \right)} \)

Explanation:
1. [x] = 1 on [1, 2), 2 on [2, 3), 3 on [3, 4).
2. ∫ = 0 + log 2 + log 3 = log 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: (c) \( 64 \)

Explanation:
1. Put x³ = t, so 3dx/x = dt/t.
2. The integral becomes ∫1^64 e^(sin t)/t dt = F(64) - F(1).
3. So k = 64.

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.