Integration Techniques (Substitution, Parts, Partial Fractions): JEE Questions & Solutions

If ∫ (sin x)^(-11/2) (cos x)^(-5/2) dx is equal to -p₁/q₁(cot x)^(9/2) -p₂/q₂(cot x)^(5/2) -p₃/q₃(cot x)^(1/2) +p₄/q₄(cot x)^(-3/2) + C,…

Let for f(x) = 7tan⁸ x + 7tan⁶ x - 3tan⁴ x - 3tan² x, I₁ = ∫₀^(π/4) f(x)dx and I₂ = ∫₀^(π/4) x f(x)dx. Then 7I₁ + 12I₂ is equal to:

Let I(x) = ∫ dx/(x-11)^(11/13) (x+15)^(15/13). If I(37) - I(24) = 1/4 ( 1/b^(1/13) - 1/c^(1/13)), where b, c ∈ ℕ, then 3(b+c) is equal to:

Let ∫ x³ sin x dx = g(x) + C, where C is the constant of integration. If 8( g( π/2) + g'( π/2)) = α π³ + β π² + γ, α, β, γ ∈ ℤ, then α + β…

Let f(x) = ∫ x³ √(3-x²) dx. If 5f(√2) = -4, then f(1) is equal to

If ∫ ( 1/x + 1/x³) ( 23-root √(3x⁻²⁴ + x⁻²⁶)) dx is equal to -α/3(α + 1) ( 3x^β + x^γ)^((α + 1)/α) + C, x > 0, where α, β, γ ∈ ℤ and C is…

Let f(x) be a positive function and I₁ = ∫_(-1/2)¹ 2x f(2x(1-2x)) dx and I₂ = ∫₋₁² f(x(1-x)) dx. Then the value of I₂/I₁ is equal to

If ∫₀¹ 1/(√(3 + x) + √(1 + x)) dx = a + b√2 + c√3, where a, b, c are rational numbers, then 2a + 3b - 4c is equal to:

The value of ∫_(π/3)^(π/2) (2 + 3 sin x)/(sin x (1 + cos x)) dx is equal to:

Let I(x) = ∫ x²(xsec² x + tan x)/(xtan x + 1)² dx. If I(0) = 0, then I(π/4) is equal to:

Let I(x) = ∫ (x+1)/x(1+x e^x)² dx, x > 0. If lim_(x → ∞) I(x) = 0, then I(1) is equal to:

Let f(x)=∫ dx/(3+4x²)√(4-3x²), |x|<2/√3. If f(0)=0 and f(1)=1/αβtan⁻¹ (α/β), α,β>0, then α²+β² is equal to