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Solution

Evaluate $$\int x^2 e^{3x}\, dx$$ using integration by parts.

Step 1: Set up IBP with $u = x^2$ and $dv = e^{3x}\,dx$.

Then $du = 2x\,dx$ and $v = \tfrac{1}{3}e^{3x}$.

Step 2: Apply the formula $\int u\,dv = uv - \int v\,du$:

$$\int x^2 e^{3x}\,dx = \frac{x^2 e^{3x}}{3} - \frac{2}{3}\int x e^{3x}\,dx$$

Step 3: Apply IBP again for $\int x e^{3x}\,dx$:

$$\int x e^{3x}\,dx = \frac{x e^{3x}}{3} - \frac{e^{3x}}{9}$$

Step 4: Combine and simplify:

$$\boxed{\int x^2 e^{3x}\,dx = \frac{e^{3x}}{27}\left(9x^2 - 6x + 2\right) + C}$$