Definite Integration

∫₂² f(x) dx = 0, ∫₁³ f(x) dx = ∫₁² f(x) dx + ∫₂³ f(x) dx, ∫₀¹ f(x) dx = ∫₀¹ f(t) dt

Property 1: Same limits give zero. Property 4: Split integration interval. Property 3: Dummy variable property. The last option is wrong due to Property 2 (sign change).

3/2

Step 1: ∫(x + 1) dx = x²/2 + x + c. Step 2: [x²/2 + x]₀¹ = (1/2 + 1) - (0 + 0) = 3/2. Split into ∫x dx + ∫1 dx for clarity.

∫₋₃³ x³ dx, ∫₋₅⁵ sin x dx, ∫₋₂² x⁵ dx

For odd functions f(-x) = -f(x), ∫₋ₐᵃ f(x) dx = 0. x³, sin x, and x⁵ are odd functions. x² is even, so its integral over symmetric limits is non-zero.