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To solve the integral \(\int x 2^x \, dx\), we can use integration by parts. 

The formula for integration by parts is:
\[
\int u \, dv = uv - \int v \, du
\]

For this integral, we choose:
\[
u = x \quad \text{and} \quad dv = 2^x \, dx
\]

Next, we need to find \(du\) and \(v\):
\[
du = dx \quad \text{and} \quad v = \int 2^x \, dx
\]

To find \(v\), recall that the integral of \(2^x\) is:
\[
\int 2^x \, dx = \frac{2^x}{\ln(2)}
\]

Now, applying integration by parts:
\[
\int x 2^x \, dx = x \cdot \frac{2^x}{\ln(2)} - \int \frac{2^x}{\ln(2)} \, dx
\]

We need to integrate \(\frac{2^x}{\ln(2)}\):
\[
\int \frac{2^x}{\ln(2)} \, dx = \frac{1}{\ln(2)} \int 2^x \, dx = \frac{1}{\ln(2)} \cdot \frac{2^x}{\ln(2)} = \frac{2^x}{(\ln(2))^2}
\]

Putting it all together:
\[
\int x 2^x \, dx = \frac{x 2^x}{\ln(2)} - \frac{2^x}{(\ln(2))^2} + C
\]

So, the solution is:
\[
\int x 2^x \, dx = \frac{x 2^x}{\ln(2)} - \frac{2^x}{(\ln(2))^2} + C
\]

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How do you integrate functions using substitution?
2. What are the steps for solving integrals using partial fractions?
3. Can you explain how to find the integral of exponential functions?
4. How does integration by parts apply to trigonometric functions?
5. What are the common techniques for solving definite integrals?

**Tip:** When solving integrals, choosing the right technique, such as substitution or integration by parts, can simplify the process significantly.

Learn how to solve the integral ∫ x 2^x dx using integration by parts. This problem involves key concepts like integration by parts and the integral of exponential functions. Suitable for college-level students.

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