Evaluate the following:

a).

b). , where

a) To evaluate the integral \( \int x^2e^{2x} \, dx \):

We can use integration by parts. The formula for integration by parts is:

\[ \int u \, dv = uv - \int v \, du \]

Let's choose:

\[ u = x^2 \]
\[ dv = e^{2x} \, dx \]

Then, we find:

\[ du = 2x \, dx \]
\[ v = \frac{1}{2} e^{2x} \]

Now, using the integration by parts formula:

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