Question - Solving Integration Problems
Solution:
The given image presents three separate integration problems, asking to integrate the following expressions with respect to x:(a) ∫(2x - 3)dx(b) ∫(2 + 3x)dx(c) ∫(x^2 - 1)dxLet's integrate each one:(a) To integrate ∫(2x - 3)dx, we integrate each term separately:∫2x dx = 2∫x dx = 2 * (1/2)x^2 = x^2∫(-3) dx = -3xTherefore, ∫(2x - 3)dx = x^2 - 3x + C, where C is the constant of integration.(b) Similarly for ∫(2 + 3x)dx:∫2 dx = 2x∫3x dx = 3 * (1/2)x^2 = (3/2)x^2Adding these together gives us ∫(2 + 3x)dx = 2x + (3/2)x^2 + C.(c) For ∫(x^2 - 1)dx:∫x^2 dx = (1/3)x^3∫(-1) dx = -xCombining these results: ∫(x^2 - 1)dx = (1/3)x^3 - x + C.So, the integrals for each part are:(a) x^2 - 3x + C(b) 2x + (3/2)x^2 + C(c) (1/3)x^3 - x + C
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