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.