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The image asks how many times integration by parts is needed to solve the integral \(\int e^{9x} \sin(34x) \, dx\).

### Step-by-Step Solution:

The integral of the product of functions like \(e^{ax} \sin(bx)\) often requires **two applications of integration by parts** due to the interaction between the exponential and trigonometric functions.

#### Integration by Parts Formula:
\[
\int u \, dv = uv - \int v \, du
\]
- Let \( u = e^{9x} \) and \( dv = \sin(34x) \, dx \).

After the first integration by parts, you'll need to apply the technique again to fully reduce the problem because the result will still contain similar terms involving both \(e^{9x}\) and \(\cos(34x)\), requiring another application.

Thus, the correct answer is **2 integrations by parts**.

Would you like to see the detailed step-by-step process for the integration?

Here are some related questions to expand your understanding:
1. How does integration by parts work for other combinations of functions, like \(e^{ax} \cos(bx)\)?
2. Why does the combination of an exponential and trigonometric function often lead to repeated integration by parts?
3. What are other methods besides integration by parts to solve such integrals?
4. Can the tabular method of integration by parts help in this scenario?
5. How would the solution differ if \( \sin(34x) \) were replaced by \( \cos(34x) \)?

**Tip:** In repeated integration by parts, watch for patterns in the resulting integrals, as they often form solvable systems.

In order to integrate ∫ e^(9x) sin(34x) dx, how many times do we need to apply integration by parts?

Find out how many times integration by parts is required to solve the integral ∫ e^(9x) sin(34x) dx. This solution covers advanced integration techniques, including the application of the integration by parts formula multiple times.

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