To evaluate the integral

$\int_1^4 \left( 8x^3 - 3x^2 + 5x + 3 \right) \, dx,$

we can use the power rule for integration, which states:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.$

Let's break the integral down term by term.

Step 1: Integrate each term

• The integral of $8x^3$ is:

$\int 8x^3 \, dx = \frac{8x^4}{4} = 2x^4.$

• The integral of $-3x^2$ is:

$\int -3x^2 \, dx = \frac{-3x^3}{3} = -x^3.$

• The integral of $5x$ is:

$\int 5x \, dx = \frac{5x^2}{2}.$

• The integral of $3$ is:

$\int 3 \, dx = 3x.$

Step 2: Evaluate the definite integral

Now, we combine all the integrals:

$\int_1^4 \left( 8x^3 - 3x^2 + 5x + 3 \right) \, dx = \left[ 2x^4 - x^3 + \frac{5x^2}{2} + 3x \right]_1^4.$

We will now evaluate this expression at the bounds $x = 4$ and $x = 1$.

Evaluate at $x = 4$:

$2(4)^4 - (4)^3 + \frac{5(4)^2}{2} + 3(4) = 2(256) - 64 + \frac{5(16)}{2} + 12 = 512 - 64 + 40 + 12 = 500.$

Evaluate at $x = 1$:

$2(1)^4 - (1)^3 + \frac{5(1)^2}{2} + 3(1) = 2(1) - 1 + \frac{5(1)}{2} + 3 = 2 - 1 + \frac{5}{2} + 3 = 4.5.$

Step 3: Subtract the values

Now, subtract the result at $x = 1$ from the result at $x = 4$:

$500 - 4.5 = 495.5.$

Thus, the value of the integral is:

$\int_1^4 \left( 8x^3 - 3x^2 + 5x + 3 \right) \, dx = 495.5.$

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Would you like a more detailed explanation of any of the steps, or have any specific questions about this integral? Here are some related questions you might find helpful:

1. What are the key rules of integration?
2. How do you handle polynomial terms when integrating?
3. How does the power rule for integration work in general?
4. What happens if you evaluate the definite integral in reverse order (i.e., from 4 to 1)?
5. What are some common mistakes when evaluating definite integrals?

Tip: Always double-check your boundary evaluations and simplify terms before subtracting for clarity!