Integrate 2 multiplied by pi to the power 3x

To integrate \( 2 \cdot \pi^{3x} \) with respect to \( x \), we follow these steps:

1. **Identify the integral:**
   \[
   \int 2 \pi^{3x} \, dx
   \]

2. **Constant multiple rule:** The constant \( 2 \) can be factored out of the integral.
   \[
   2 \int \pi^{3x} \, dx
   \]

3. **Substitution:** Let \( u = 3x \), hence \( du = 3 \, dx \) or \( dx = \frac{du}{3} \).

4. **Rewrite the integral in terms of \( u \):**
   \[
   2 \int \pi^u \cdot \frac{du}{3}
   \]
   Simplify the constants:
   \[
   \frac{2}{3} \int \pi^u \, du
   \]

5. **Integrate \( \pi^u \) with respect to \( u \):**
   The integral of \( \pi^u \) with respect to \( u \) is \(\frac{\pi^u}{\ln(\pi)}\), because \(\int a^x \, dx = \frac{a^x}{\ln(a)}\) for any constant \( a \).

6. **Apply the result of the integration:**
   \[
   \frac{2}{3} \cdot \frac{\pi^u}{\ln(\pi)}
   \]

7. **Substitute \( u = 3x \) back into the integral:**
   \[
   \frac{2}{3} \cdot \frac{\pi^{3x}}{\ln(\pi)}
   \]

8. **Combine the constants:**
   \[
   \frac{2}{3 \ln(\pi)} \pi^{3x}
   \]

Thus, the integral is:
\[
\int 2 \pi^{3x} \, dx = \frac{2 \pi^{3x}}{3 \ln(\pi)} + C
\]
where \( C \) is the constant of integration.

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### Related Questions
1. What is the integral of \( \pi^{kx} \) for a constant \( k \)?
2. How do you find the derivative of \( \pi^{3x} \)?
3. What is the integral of \( e^{kx} \) for a constant \( k \)?
4. How do you integrate exponential functions with different bases?
5. What are the properties of the natural logarithm function used in integration?
6. Can you integrate \( 2 \cdot e^{3x} \) and compare it to the integral of \( 2 \cdot \pi^{3x} \)?
7. How do you approach integrals involving trigonometric functions raised to a variable power?
8. What are some real-life applications of integrating exponential functions?

### Tip
When integrating functions of the form \( a^{kx} \), remember that the result involves dividing by the natural logarithm of the base \( a \).

Learn how to integrate the function 2 multiplied by pi to the power 3x step-by-step. This problem involves the integration of exponential functions and applies the integral formula for exponential functions. Suitable for advanced high school students and beyond.

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