Z
−3x
−2 −
3
x
4
+
2
x
+ 5x
0
(b) dx

It seems like you're asking for the integration of the expression. Let me try to simplify and interpret the given expression for clarity before proceeding with integration.

The expression appears to be:

\[
Z - 3x - 2 - \frac{3x}{4} + 2x + 5x^0
\]

- \( Z \) seems like a constant.
- \( 3x \), \( 2x \), and \( \frac{3x}{4} \) are linear terms involving \( x \).
- \( 5x^0 \) is a constant, because \( x^0 = 1 \).

Now, simplifying the expression:

\[
Z - 3x - 2 - \frac{3x}{4} + 2x + 5
\]

We can combine like terms:

- Combine \( -3x \), \( \frac{-3x}{4} \), and \( 2x \):
  \[
  -3x + 2x = -x, \quad -x - \frac{3x}{4} = -\frac{7x}{4}
  \]

So the simplified expression becomes:

\[
Z - \frac{7x}{4} + 3
\]

Now, integrating with respect to \( x \):

\[
\int \left(Z - \frac{7x}{4} + 3 \right) \, dx
\]

The integral is:

\[
Zx - \frac{7x^2}{8} + 3x + C
\]

where \( C \) is the constant of integration.

Would you like more details on any part of this integration, or have any other questions?

Here are 5 related questions to explore further:

1. How do you integrate expressions with fractions?
2. What is the process of combining like terms in algebraic expressions?
3. How do you handle integration when there are multiple constants involved?
4. Can you explain the importance of the constant of integration in indefinite integrals?
5. How would the integration change if the expression included higher powers of \( x \)?

**Tip:** Always simplify the expression first before attempting to integrate, as it makes the process smoother.

Explore the step-by-step solution for simplifying and integrating the algebraic expression Z - 3x - 2 - 3x/4 + 2x + 5. This problem involves combining like terms and performing integration with respect to x, suitable for students in Grades 11-12 or early college calculus.

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