Setting k(1) + 1 = 2(1) - 1 gives k + 1 = 1, so k = 0.

Setting k(1) + 1 = 3 gives k = 2 for continuity.

Setting k(2) + 1 = 2^2 - 3 gives 2k + 1 = 1, leading to k = 0.

Setting k(1) + 2 = 3 gives k = 1.

Setting k = 0 for continuity at x = 0 gives f(0) = 0.

Setting k(0) = 0^2 + 1 gives k = 1.

Setting 2(1) + m = m(1) + 3 gives m = 1.

Setting 2(1) + m = 1^2 + 1 gives m = 0.

Setting the two pieces equal at x = 3 gives us 6 + m = 6. Thus, m = 0.

Setting 3(1) + m = 2(1)^2 gives m = -1.

Setting the two pieces equal at x = 1: 1^2 + m = 4 - 1. Solving gives m = 2.

Setting x^2 + m = mx + 1 at x = 1 gives m = 1 for continuity.

f'(x) = 4x^3 - 12x^2 + 12x. Evaluating at x = 1 gives f'(1) = 4 - 12 + 12 = 4.

The integral evaluates to [x - (1/3)x^3] from 0 to 1 = 1 - 1/3 = 2/3.

The integral ∫(3x^2)dx from 0 to 1 = [x^3] from 0 to 1 = 1.

The integral evaluates to [(1/3)x^3 + x^2] from 0 to 1 = (1/3 + 1) - 0 = 4/3.

The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.

The integral evaluates to [e^x] from 0 to 1 = e - 1.

Using integration by parts, the integral evaluates to (e - 1)/2.

The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.

The integral evaluates to [x^3/3 + (3/2)x^2 + 2x] from 0 to 1 = (1/3 + 3/2 + 2) - 0 = 4/3 + 3/2 = 2.5.

The integral evaluates to [x^3/3 - x^2 + x] from 0 to 1 = (1/3 - 1 + 1) = 1/3.

The integral evaluates to [x^4/4 - x^3 + 2x] from 0 to 1 = (1/4 - 1 + 2) - (0) = 1/4.

The integral evaluates to [x^4/4 - x^3 + (3/2)x^2 - x] from 0 to 1 = 0.

The integral evaluates to [x^4/4 - x^3 + (3/2)x^2] from 0 to 1 = (1/4 - 1 + 3/2) = 1/4.

The integral evaluates to [x^4/4 - 2x^2 + 4x] from 0 to 1 = (1/4 - 2 + 4) = (1/4 + 2) = 9/4.

The integral evaluates to [x^5/5 + (2/3)x^3] from 0 to 1 = (1/5 + 2/3) = 11/15.

The integral evaluates to [x^5/5 - x^4 + 2x^3 - 2x^2 + x] from 0 to 1 = (1/5 - 1 + 2 - 2 + 1) = 0.

The integral evaluates to [x^5/5] from 0 to 1 = 1/5.

The integral evaluates to [x^3/3 - x^2 + x] from 0 to 2 = (8/3 - 4 + 2) = 2/3.