To solve the integral $\int xe^{2x} \, dx$, we can use the method of integration by parts. Integration by parts is based on the formula:

$\int u \, dv = uv - \int v \, du$

For the integral $\int xe^{2x} \, dx$, let's choose:

• $u = x$ (so that $du = dx$)
• $dv = e^{2x} \, dx$ (so that $v = \frac{1}{2}e^{2x}$)

Now, apply the integration by parts formula:

$\int xe^{2x} \, dx = x \cdot \frac{1}{2}e^{2x} - \int \frac{1}{2}e^{2x} \, dx$

Simplify the integral:

$\int xe^{2x} \, dx = \frac{1}{2}xe^{2x} - \frac{1}{2} \int e^{2x} \, dx$

The integral $\int e^{2x} \, dx$ is straightforward:

$\int e^{2x} \, dx = \frac{1}{2}e^{2x}$

So the expression becomes:

$\int xe^{2x} \, dx = \frac{1}{2}xe^{2x} - \frac{1}{2} \cdot \frac{1}{2}e^{2x} + C$

Simplifying further:

$\int xe^{2x} \, dx = \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} + C$

where $C$ is the constant of integration.

So, the final answer is:

$\int xe^{2x} \, dx = \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} + C$

Would you like more details on this process or have any questions?

Here are 5 related questions:

1. How would you apply integration by parts to $\int x^2 e^{2x} \, dx$?
2. What is the integral of $xe^{-2x} \, dx$?
3. Can you solve $\int x \sin(2x) \, dx$ using integration by parts?
4. How would the solution change if the exponent was $3x$ instead of $2x$?
5. What is the general formula for integrating $x e^{ax} \, dx$?

Tip: When using integration by parts, always carefully choose $u$ and $dv$ to simplify the integral you're trying to solve.