This integral problem involves evaluating:

$\int_{-1}^{5} (x^2 + 4x) \, dx$

Solution:

1. Find the antiderivative of $x^2 + 4x$:

   • The antiderivative of $x^2$ is $\frac{x^3}{3}$.
   • The antiderivative of $4x$ is $2x^2$.

   So, the antiderivative of $x^2 + 4x$ is: $F(x) = \frac{x^3}{3} + 2x^2$

2. Evaluate the definite integral: Substitute the limits of integration ($x = -1$ and $x = 5$) into $F(x)$: $\int_{-1}^{5} (x^2 + 4x) \, dx = F(5) - F(-1)$

   Compute $F(5)$: $F(5) = \frac{5^3}{3} + 2(5^2) = \frac{125}{3} + 50 = \frac{125}{3} + \frac{150}{3} = \frac{275}{3}$

   Compute $F(-1)$: $F(-1) = \frac{(-1)^3}{3} + 2(-1)^2 = \frac{-1}{3} + 2 = \frac{-1}{3} + \frac{6}{3} = \frac{5}{3}$

3. Subtract $F(-1)$ from $F(5)$: $F(5) - F(-1) = \frac{275}{3} - \frac{5}{3} = \frac{270}{3} = 90$

Final Answer:

$\int_{-1}^{5} (x^2 + 4x) \, dx = 90$

Would you like further explanations or steps?

Here are 5 related questions for practice:

1. What is the integral of $x^3 + 6x^2$ from 0 to 4?
2. How do you compute an indefinite integral of $2x^2 + 3x$?
3. What is the geometric interpretation of a definite integral?
4. How does changing the limits of integration affect the result of a definite integral?
5. What are the properties of definite integrals (e.g., linearity)?

Tip: Always check the continuity of the function over the integration interval before evaluating a definite integral!