To solve the problem, let's evaluate each definite integral separately and then subtract the results as specified.

Problem:

Evaluate: $\int_{2}^{4} (x^2 + 3x + 1) \, dx - \int_{2}^{4} (2x^2 - 7x + 1) \, dx$

Step 1: Evaluate $\int_{2}^{4} (x^2 + 3x + 1) \, dx$

1. Find the antiderivative of $x^2 + 3x + 1$: $\int (x^2 + 3x + 1) \, dx = \frac{x^3}{3} + \frac{3x^2}{2} + x$

2. Apply the limits of integration from 2 to 4: $\left[ \frac{x^3}{3} + \frac{3x^2}{2} + x \right]_{2}^{4}$

   • At $x = 4$: $\frac{4^3}{3} + \frac{3 \cdot 4^2}{2} + 4 = \frac{64}{3} + \frac{48}{2} + 4 = \frac{64}{3} + 24 + 4 = \frac{64}{3} + 28 = \frac{148}{3}$
   • At $x = 2$: $\frac{2^3}{3} + \frac{3 \cdot 2^2}{2} + 2 = \frac{8}{3} + \frac{12}{2} + 2 = \frac{8}{3} + 6 + 2 = \frac{8}{3} + 8 = \frac{32}{3}$

3. Subtract the two results: $\frac{148}{3} - \frac{32}{3} = \frac{116}{3}$

Step 2: Evaluate $\int_{2}^{4} (2x^2 - 7x + 1) \, dx$

1. Find the antiderivative of $2x^2 - 7x + 1$: $\int (2x^2 - 7x + 1) \, dx = \frac{2x^3}{3} - \frac{7x^2}{2} + x$

2. Apply the limits of integration from 2 to 4: $\left[ \frac{2x^3}{3} - \frac{7x^2}{2} + x \right]_{2}^{4}$

   • At $x = 4$: $\frac{2 \cdot 4^3}{3} - \frac{7 \cdot 4^2}{2} + 4 = \frac{128}{3} - \frac{112}{2} + 4 = \frac{128}{3} - 56 + 4 = \frac{128}{3} - 52 = \frac{-28}{3}$
   • At $x = 2$: $\frac{2 \cdot 2^3}{3} - \frac{7 \cdot 2^2}{2} + 2 = \frac{16}{3} - \frac{28}{2} + 2 = \frac{16}{3} - 14 + 2 = \frac{16}{3} - 12 = \frac{-20}{3}$

3. Subtract the two results: $\frac{-28}{3} - \frac{-20}{3} = \frac{-8}{3}$

Step 3: Subtract the Integrals

Combine the results from Step 1 and Step 2: $\frac{116}{3} - \left(\frac{-8}{3}\right) = \frac{116}{3} + \frac{8}{3} = \frac{124}{3}$

Final Answer:

$\int_{2}^{4} (x^2 + 3x + 1) \, dx - \int_{2}^{4} (2x^2 - 7x + 1) \, dx = \frac{124}{3}$

Would you like more details on any part of the solution?

Here are some questions to further explore related topics:

1. What is the process for finding antiderivatives for polynomial functions?
2. How do definite integrals differ from indefinite integrals?
3. Can definite integrals be interpreted in terms of area under a curve?
4. How would you handle definite integrals with trigonometric functions?
5. What are some common techniques for evaluating integrals without direct computation?

Tip: Always double-check limits of integration, as reversing them changes the sign of the result.