Properties of Definite Integrals: JEE Questions & Solutions

The value of ∫_(-π/6)^(π/6) (π+4x¹¹)/(1-sin(|x|+π/6)) dx is equal to:

The number of elements in the set S = { x: x ∈ [0, 100] and ∫₀^x t² sin(x-t) dt = x² } is

In I(m, n) = ∫₀¹ x^(m-1)(1-x)ⁿ⁻¹ dx, m, n > 0, then I(9, 14) + I(10, 13) is:

If ∫_(-π/2)^(π/2) (96x² cos² x)/(1 + e^x) dx = π(aπ² + β), a, β ∈ ℤ, then (a + β)² equals:

The integral ∫₀^π (8x dx)/(4 cos² x + sin² x) is equal to

The integral ∫₀^π ((x + 3) sin x)/(1 + 3 cos² x) dx is equal to:

Let f(x) = ∫₀^x g(t) log((1 - t)/(1 + t)) dt, where g is a continuous odd function. If ∫_(-π/2)^(π/2) ((f(x) x² cos(x))/(1 + e^x)) dx =…

Let f: ℝ → ℝ be a function defined by f(x) = 4^x/(4^x + 2) and M = ∫_f(a)^f(1-a) x sin⁴(x(1 - x)) dx, N = ∫_f(a)^f(1-a) sin⁴(x(1 - x)) dx;…

The value of ∫₀¹ (2x³ - 3x² - x + 1)^(1/3) dx is equal to:

The value of k ∈ ℕ for which the integral Iₙ = ∫₀¹ (1 - x^k)ⁿ dx, n ∈ ℕ, satisfies 147I₂₀ = 148I₂₁, is:

The value of the integral ∫_(-π/4)^(π/4) (x + π/4)/(2 - cos 2x) dx is:

If ∫₀^π (5^(cos x) (1 + cos x cos 3x + cos² x + cos³ x cos 3x))/(1 + 5^(cos x)) dx = (k π)/16, then k is equal to.

Let f(x) be a function satisfying f(x) + f(π - x) = π², for all x ∈ ℝ. Then ∫₀^π f(x)sin x dx is equal to:

For m, n > 0, let α(m, n) = ∫₀² t^m (1 + 3t)ⁿ dt. If 11 α(10, 6) + 18 α(11, 5) = p (14)⁶, then p is equal to:

If f: ℝ → ℝ is a continuous function satisfying ∫₀^(π/2) f(sin 2x) sin x dx + α ∫₀^(π/4) f(cos 2x) cos x dx = 0, then the value of α is:

If ∫₀¹1/(5+2x-2x²)(1+e^(2-4x)) dx =1/αlogₑ ((α+1)/β), α,β>0, then α⁴-β⁴ is equal to