To solve \(\int_{0}^{1} xe^{2x} \, dx\), we use integration by parts. Let \(u = x\) and \(dv = e^{2x} \, dx\).

Then \(du = dx\) and \(v = \frac{1}{2}e^{2x}\).

Using integration by parts, \(\int u \, dv = uv - \int v \, du\), we have:

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

\(= \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} + C\).

Now, evaluate from 0 to 1:

\(\left[ \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} \right]_{0}^{1}\).

At \(x = 1\): \(\frac{1}{2} \cdot 1 \cdot e^{2} - \frac{1}{4}e^{2} = \frac{1}{4}e^{2}\).

At \(x = 0\): \(\frac{1}{2} \cdot 0 \cdot e^{0} - \frac{1}{4}e^{0} = -\frac{1}{4}\).

Thus, the definite integral is:

\(\frac{1}{4}e^{2} - (-\frac{1}{4}) = \frac{1}{4}(e^{2} + 1)\).