1) The integral of dx/(sin^2x + cos^2x) simplifies to dx/1 which is just dx. Therefore, the result is ∫ dx = x + C where C is the constant of integration.

2) To integrate x √(1 - 8x^2) × dx, let u = 1 - 8x^2. Then, du = -16xdx.
Rearranging, we get -1/16 du = x dx.
Therefore, the integral becomes -(1/16) ∫ √u du.
This simplifies to -(1/16) (2/3) u^(3/2) + C = -(1/24) * (1 - 8x^2)^(3/2) + C.

3) To integrate 3xe^(2x) × dx, use integration by parts. Let u = 3x and dv = e^(2x) dx. Then, du = 3 dx and v = (1/2)e^(2x).
Apply the integration by parts formula: ∫ udv = uv - ∫ vdu.
Therefore, the integral becomes 3xe^(2x)/2 - 3/2 ∫ e^(2x) dx.
This simplifies to (3x/2) e^(2x) - (3/4) e^(2x) + C = (3x/2 - 3/4) e^(2x) + C.

4) To integrate (e^x dx)/√(1 + e^x), perform the substitution of u = 1 + e^x. Then, du = e^x dx.
The integral becomes ∫du/√u.
This simplifies to 2√u + C = 2√(1 + e^x) + C.