Indefinite Integrals: JEE Questions & Solutions

If ∫₀¹ 4 cot⁻¹(1 - 2x + 4x²) dx = a tan⁻¹(2) - b ln(5), where a, b ∈ ℕ, then (2a + b) is equal to.

Let f(x) = ∫ ((2 - x²) · e^x)/(√(1 + x))(1 - x)^(3/2) dx. If f(0) = 0, then f(1/2) is equal to:

Let I(x) = ∫ 3dx/(4x+6)√(4x²+8x+3) and I(0) = √3/4 + 20. If I(1/2) = a√2/b + c, where a, b, c ∈ ℕ, gcd(a, b) = 1, then a+b+c is equal to

Let f(t)=∫ ((1-sin(logₑ t))/(1-cos(logₑ t)))dt, t>1. If f(e^(π/2))=-e^(π/2) and f(e^(π/4))=α e^(π/4), then α equals

If f(x) satisfies the relation f(x)=e^x+∫₀¹ (y+x e^x)f(y) dy, then e+f(0) is equal to

If ∫ (1-5cos² x)/(sin⁵ xcos² x) dx=f(x)+C, where C is the constant of integration, then f (π/6)-f (π/4) is equal to:

Let f(x)=∫ dx/(x^(2/3)+2x^(1/2)) such that f(0)=-26+24logₑ(2). If f(1)=a+blogₑ(3), where a,b∈ℤ, then a+b is equal to:

If ∫ e^x( (x sin⁻¹ x)/√(1 - x²) + (sin⁻¹ x)/(1 - x²)^(3/2) + x/(1 - x²)) dx = g(x) + C, where C is the constant of integration, then g(1/2)…

If ∫ (2x² + 5x + 9)/√(x² + x + 1) dx = x√(x² + x + 1) + α √(x² + x + 1) + β logₑ ( | x + 1/2 + √(x² + x + 1) |) + C, where C is the…

Let f be a real-valued continuous function defined on the positive real axis such that g(x) = ∫₀^x t f(t) dt. If g(x³) = x⁶ + x⁷, then the…

If f(x) = ∫ 1/(x^(1/4) (1 + x^(1/4))) dx, f(0) = -6, then f(1) is equal to:

If 24 ∫₀^(π/4) ( sin|4x - π/12| + [2 sin x]) dx = 2π + α, where [ · ] denotes the greatest integer function, then α is equal to:

If ∫ ( √(1 + x²) + x)¹⁰/( √(1 + x²) - x)⁹ dx = 1/m ( ( √(1 + x²) + x)ⁿ ( n√(1 + x²) - x)) + C, where m, n ∈ ℕ and C is the constant of…

The integral ∫ (x⁸ - x²)/(x¹² + 3x⁶ + 1) arctan(x³ + 1/x³) dx is equal to:

If ∫ (sin^(3/2)x + cos^(3/2)x)/√(sin³ x cos³ x sin(x-θ)) dx = A√(cosθ tan x - sinθ) + B√(cosθ - sinθ cot x) + C, where C is the integration…

Let f(x) = ∫₋ₓ^x (|t| - t²)e^(-t²) dt and g(x) = ∫₀^x² t^(1/2)e^(-t) dt. The value of f(√(ln 9)) + g(√(ln 9)) is:

The minimum value of the function f(x) = ∫₀² e^(|x-t|) dt is:

Let f(x) = ∫ 2x/(x² + 1)(x² + 3) dx. If f(3) = 1/2(logₑ 5 - logₑ 6), then f(4) is equal to:

If ∫ √(sec 2x - 1) dx = α logₑ | cos 2x + β + √(cos 2x ( 1 + cos 1/β x)) | + constant then β - α is equal to

Let α ∈ (0, 1) and β = log(1 - α). Let Pₙ(x) = x + x²/2 + x³/3 + … + xⁿ/n, x ∈ (0, 1). Then the integral ∫₀^α t⁵⁰/(1 - t) dt is equal to:

Let α > 0. If ∫₀^α x/(√(x + α) - √x) dx = (16 + 20√2)/15, then α is equal to:

If I(x) = ∫ e^sin²x(cos x sin 2x - sin x) dx and I(0) = 1, then I(π/3) is equal to:

For α, β, γ, δ ∈ ℕ, if ∫ ( x/e)^2x + ( e/x)^2x log x dx = 1/α ( x/e)^(β x) - 1/γ ( e/x)^(δ x) + C where e = Σ_(n=0)^∞ 1/n! and C is the…

Let I(x) = ∫ √((x+7)/x) dx and I(9) = 12 + 7 logₑ 7. If I(1) = α + 7 logₑ (1+2√2), then α⁴ is equal to: