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We are given the following equation to solve:

\[
\int (2x - 1) e^{2x+3} \, dx = yz - \int z \, dy
\]
and the task is to find \(\int z \, dy\).

### Step 1: Solving the left side \(\int (2x - 1) e^{2x+3} \, dx\)
To solve this, we will use integration by parts.

#### Let:
- \( u = 2x - 1 \), hence \( du = 2 \, dx \).
- \( dv = e^{2x+3} \, dx \), hence \( v = \frac{1}{2} e^{2x+3} \) (since the derivative of \(2x + 3\) is 2).

Now apply integration by parts:
\[
\int u \, dv = u v - \int v \, du
\]
Substituting the values:
\[
\int (2x - 1) e^{2x+3} \, dx = (2x - 1) \cdot \frac{1}{2} e^{2x+3} - \int \frac{1}{2} e^{2x+3} \cdot 2 \, dx
\]
This simplifies to:
\[
\frac{1}{2} (2x - 1) e^{2x+3} - \int e^{2x+3} \, dx
\]
The integral of \(e^{2x+3}\) is:
\[
\frac{1}{2} e^{2x+3}
\]
So, the result becomes:
\[
\frac{1}{2} (2x - 1) e^{2x+3} - \frac{1}{2} e^{2x+3}
\]
Simplifying further:
\[
\frac{1}{2} e^{2x+3} [(2x - 1) - 1] = \frac{1}{2} e^{2x+3} (2x - 2) = e^{2x+3} (x - 1)
\]

### Step 2: Rearranging the equation
We are given that:
\[
\int (2x - 1) e^{2x+3} \, dx = yz - \int z \, dy
\]
From step 1, we know:
\[
e^{2x+3} (x - 1) = yz - \int z \, dy
\]
Rearranging to find \(\int z \, dy\):
\[
\int z \, dy = yz - e^{2x+3} (x - 1)
\]

### Step 3: Comparing with the given options
Looking at the structure, the closest match to \(\int z \, dy\) from the options appears to be:

\[
\boxed{\frac{1}{2} e^{2x+3} + c}
\]

which matches option (b).

---

Would you like further clarification on any steps or have additional questions? Here are some questions you might find helpful:

1. What is integration by parts, and when is it used?
2. How do we determine the choice of \( u \) and \( dv \) in integration by parts?
3. What is the integral of an exponential function like \( e^{2x+3} \)?
4. How can you verify the solution to a definite integral?
5. What are the conditions for applying the method of substitution in integrals?

**Tip:** For solving complex integrals, it's important to identify which method (substitution, integration by parts, etc.) will simplify the expression the most effectively.

If ∫(2x - 1)e^(2x + 3) dx = yz - ∫z dy, then ∫z dy = ...

Solve an integral using the integration by parts method for the equation ∫(2x - 1)e^(2x + 3) dx. Learn how to apply the technique to handle exponential functions and their derivatives. Suitable for university-level calculus students.

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