Solving Integration Problems
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)dx
Let'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 = -3x
Therefore, ∫(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^2
Adding 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 = -x
Combining 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
