Evaluate \int_1^4 (5 - 2x + 3x^2) dx :

Solution:
1. Identify the function to integrate: f(x) = 5 - 2x + 3x^2

2. Integrate each term:
- The integral of a constant 5 with respect to x is 5x.
- The integral of -2x is -\frac{2}{2}x^2 = -x^2.
- The integral of 3x^2 is \frac{3}{3}x^3 = x^3.

3. Combine the integrals:
\int (5 - 2x + 3x^2) \, dx = 5x - x^2 + x^3 + C

4. Evaluate the definite integral from 1 to 4:
- Compute the antiderivative at 4:
F(4) = 5(4) - (4)^2 + (4)^3 = 20 - 16 + 64 = 68
- Compute the antiderivative at 1:
F(1) = 5(1) - (1)^2 + (1)^3 = 5 - 1 + 1 = 5

5. Find the difference F(4) - F(1):
68 - 5 = 63

6. The value of the integral is:
\int_1^4 (5 - 2x + 3x^2) \, dx = 63