Let's break down the process of integrating the function $f(x) = -4x^3 + \frac{3}{2}x^2 - 8x + 7$, step by step.

Step 1: Basic Rules of Integration

The integral of a function is the reverse process of differentiation. For a basic power function of the form $x^n$, the rule is:

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

Where $C$ is the constant of integration. We will apply this rule to each term in the function.

Step 2: Integrating $-4x^3$

Let's start by integrating the first term, $-4x^3$.

Using the power rule:

$\int -4x^3 \, dx = -4 \cdot \frac{x^{3+1}}{3+1} = -4 \cdot \frac{x^4}{4}$

This simplifies to:

$-\frac{4}{4}x^4 = -x^4$

Step 3: Integrating $\frac{3}{2}x^2$

Next, we integrate the second term, $\frac{3}{2}x^2$.

Applying the power rule:

$\int \frac{3}{2}x^2 \, dx = \frac{3}{2} \cdot \frac{x^{2+1}}{2+1} = \frac{3}{2} \cdot \frac{x^3}{3}$

This simplifies to:

$\frac{3}{6}x^3 = \frac{1}{2}x^3$

Step 4: Integrating $-8x$

Now, we integrate the third term, $-8x$.

$\int -8x \, dx = -8 \cdot \frac{x^{1+1}}{1+1} = -8 \cdot \frac{x^2}{2}$

This simplifies to:

$-4x^2$

Step 5: Integrating $7$

Finally, we integrate the constant $7$.

The integral of a constant $c$ is simply $cx$, so:

$\int 7 \, dx = 7x$

Step 6: Combine the Results

Now, combine the integrals of all the terms:

$\int f(x) \, dx = -x^4 + \frac{1}{2}x^3 - 4x^2 + 7x + C$

Recap of the Integrated Function

The integrated function is:

$\int (-4x^3 + \frac{3}{2}x^2 - 8x + 7) \, dx = -x^4 + \frac{1}{2}x^3 - 4x^2 + 7x + C$

Would you like any clarification on these steps, or do you want to ask about any specific parts of the process?

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Here are five questions you could explore next:

1. How does the power rule of integration work with negative exponents?
2. How would you integrate a trigonometric function, like $\sin(x)$ or $\cos(x)$?
3. What happens when you integrate a function that includes a square root, such as $\sqrt{x}$?
4. How does the process differ when dealing with definite integrals?
5. What are common mistakes to avoid when applying the power rule?

Tip: Always remember to add the constant of integration $C$ when solving indefinite integrals!