Definite Integrals: JEE Questions & Solutions

6 ∫₀^π |sin 3x + sin 2x + sin x| dx is equal to:

Let [ · ] denote the greatest integer function and f(x) = lim_(n → ∞) 1/n³ Σ_(k=1)ⁿ [ k²/3^x ]. Then 12 Σ_(j=1)^∞ f(j) is equal to.

The value of the integral ∫_(-π/2)^(π/2) 1/([x]+4) dx, where [·] denotes the greatest integer function, is

Let [ ] be the greatest integer function. If α=∫₀⁶⁴(x^(1/3)-[x^(1/3)]) dx, then 1/π∫₀^απsin²θ/(sin⁶θ+cos⁶θ) dθ is equal to.

The value of the integral ∫_(π/24)^(5π/24) dx/(1 + 3-root √(tan 2x)) is:

Let a differentiable function f satisfy ∫₀³⁶ f (tx/36)dt=4α f(x). If y=f(x) is a standard parabola passing through the points (2,1) and…

The value of Σ_(r=1)²⁰√|π(∫₀^r x|sin π x| dx)| is:

Let [ · ] denote the greatest integer function. Then ∫_(-π/2)^(π/2) 12(3+[x])/(3+[sin x]+[cos x]) dx is equal to:

The value of ∫_e²^e⁴ 1/x ( e^( (logₑ x)² +1)⁻¹/(e^( (logₑ x)² +1)⁻¹ + e^( (6-logₑ x)² +1)⁻¹)) dx is:

If I = ∫₀^(π/2) (sin^(3/2) x)/(sin^(3/2) x + cos^(3/2) x) dx, then ∫₀^2I (x sin x cos x)/(sin⁴ x + cos⁴ x) dx equals:

Let f: ℝ → ℝ be a twice-differentiable function such that f(2) = 1. If F(x) = x f(x) for all x ∈ ℝ, and the integrals ∫₀² x F'(x) dx = 6…

The integral 80 ∫₀^(π/4) (sin θ + cos θ)/(9 + 16 sin 2θ) dθ is equal to:

Let f(x) = ∫₀^x t(t² - 9t + 20) dt, 1 ≤ x ≤ 5. If the range of f is [α, β], then 4(α + β) equals:

Let (a, b) be the point of intersection of the curve x² = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral…

4∫₀¹ ( 1/(√(3+x²) + √(1+x²))) dx - 3logₑ(√3) is equal to:

The value of ∫₋₁¹ (1 + √(|x| - x))e^x + (√(|x| - x))e^(-x)/(e^x + e^(-x)) dx is equal to

Let f(x) + 2f( 1/x) = x² + 5 and 2g(x) - 3g( 1/2) = x, x > 0. If α = ∫₁² f(x) dx, β = ∫₁² g(x) dx, then the value of 9α + β is:

The integral ∫₋₁^(3/2) | π² x sin(π x) | dx is equal to:

If (a,b) is the orthocentre of the triangle whose vertices are (1,2), (2,3) and (3,1), and I₁=∫ₐ^b xsin(4x-x²)dx, I₂=∫ₐ^b sin(4x-x²)dx,…

For 0 < a < 1, find the value of the integral ∫₀^π dx/(1 - 2a cos(x) + a²):

For x ∈ (-π/2, π/2), if y(x) = ∫ (csc x + sin x)/(csc x sec x + tan x sin² x) dx and lim_(x → (π/2)⁻) y(x) = 0, then y(π/4) is equal to:

If ∫_(π/6)^(π/3) √(1 - sin 2x) dx = α + β√2 + γ√3, where α, β, γ are rational numbers, then 3α + 4β - γ is equal to:

The value of lim_n→∞ (Σ_(k=1)ⁿ n³/(n²+k²)(n²+3k²)) is:

Let y = f(x) be a thrice differentiable function in (-5, 5). Let the tangents to the curve y = f(x) at (1, f(1)) and (3, f(3)) make angles…

Let f: ℝ → ℝ be defined by f(x) = ae^2x + be^x + cx. If f(0) = -1, f'(logₑ 2) = 21, and ∫₀^(logₑ 4) (f(x) - cx) dx = 39/2, then the value…

If 525∫₀^(π/2) sin 2x cos^(11/2) x (1 + cos^(5/2) x)^(1/2) dx = (n√2 - 64), then n is equal to:

Let f(x) = -2, -2 ≤ x ≤ 0 x - 2, 0 < x ≤ 2 and h(x) = f(|x|) + |f(x)|. Then ∫₋₂² h(x) dx is equal to:

The value of 9 ∫₀⁹ ⌊ √(10x/(x+1)) ⌋ dx, where ⌊ t ⌋ denotes the greatest integer less than or equal to t, is:

The integral ∫_(1/4)^(3/4) cos(2cot⁻¹√(1-x)/(1+x)) dx is equal to:

Evaluate the following limit: lim_(x → π/2) (∫_x³^(π/2)³ (sin(2t^(1/3)) + cos(t^(1/3))) dt)/(x - π/2)²

The value of the integral: ∫_(3√2/4)^(3√3/4) 48/√(9 - 4x²) dx is equal to:

The integral 16∫₁² dx/x³(x²+2)² is equal to:

If ∫_(1/3)³ |ln(x)| dx = m/n ln(n²/e), where m and n are coprime natural numbers, then m² + n² - 5 is equal to.

Let f(x) = x + a/(π² - 4) sin x + b/(π² - 4) cos x, x ∈ ℝ be a function which satisfies f(x) = x + ∫₀^(π/2) sin(x + y) f(y) dy. Then (a+b)…

Let [x] denote the greatest integer ≤ x. Consider the function f(x) = max{x², 1 + [x]}. Then the value of the integral ∫₀² f(x) dx is:

The value of the integral ∫₁² (t⁴ + 1)/(t⁶ + 1) dt is:

The value of the integral ∫_(1/2)² (tan⁻¹ x)/x dx is equal to:

If [t] denotes the greatest integer ≤ t, then the value of 3(e-1)²/e ∫₁² x² e^([x] + [x³]) dx is:

If φ(x) = 1/√x ∫_(π/4)^x ( 4√2 sin t - 3 φ'(t)) dt, x > 0, then φ'( π/4) is equal to:

If ∫₀¹ (x²¹ + x⁴ + x⁷)(2x¹⁴ + 3x⁷ + 6)^(1/7) dx = 1/7 (11)^(m/n) where l, m, n ∈ ℕ, and m, n are coprime, then l + m + n is equal to:

Let 5f(x) + 4f(1/x) = 1/x + 3, x>0. Then 18∫₁² f(x) dx is equal to:

Let [t] denote the greatest integer ≤ t. Then 2/π∫_(π/6)^(5π/6) (8[csc x] - 5[cot x]) dx is equal to:

Let f be a continuous function satisfying ∫₀^t² ( f(x) + x²) dx = 4/3 t³, ∀ t > 0. Then f ( π²/4) is equal to:

The value of the integral ∫_(-logₑ 2)^(logₑ 2) e^x ( log ( e^x + √(1 + e^2x))) dx is equal to:

Let the function f: [0, 2] → ℝ be defined as: f(x) = e^min{x², x - ⌊ x ⌋}, x ∈ [0, 1), e^(⌊ x - logₑ x ⌋), x ∈ [1, 2), where ⌊ t ⌋ denotes…

If ∫_(-0.15)^0.15 | 100x² - 1 | dx = k/3000, then k is equal to:

Evaluate: ∫₀^∞ 6/(e^3x+6e^2x+11e^x+6) dx

Let for x ∈ ℝ, S₀(x) = x, S_k(x) = C_k x + k ∫₀^x S_(k-1)(t) dt where C₀ = 1, C_k = 1 - ∫₀¹ S_(k-1)(x) dx, k = 1, 2, 3, …. Then S₂(3) + 6C₃…

The value of (e^(-π/4) + ∫₀^(π/4) e^(-x)tan⁵⁰x dx)/(∫₀^(π/4) e^(-x)(tan⁴⁹x + tan⁵¹x) dx) is:

Let fₙ = ∫₀^(π/2) ( Σ_(k=1)ⁿ sin^(k-1)x) ( Σ_(k=1)ⁿ (2k-1)sin^(k-1)x) cos x dx, n ∈ ℕ. Then f₂₁ - f₂₀ is equal to.