To solve $\int 3x e^{x^2} \, dx$ using integration by parts, follow these steps:

Integration by Parts Formula:

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

Step 1: Let $u$ and $dv$:

• Choose $u = x$ (since it simplifies upon differentiation).
• Then $dv = 3e^{x^2} dx$.

Step 2: Compute $du$ and $v$:

• Differentiate $u$: $du = dx$.
• Integrate $dv$: Since $dv = 3e^{x^2} dx$, we notice that $e^{x^2}$ requires substitution for integration. Let $w = x^2$, so $dw = 2x dx$, and rewrite: $v = \int 3e^{x^2} dx = \int 3e^w \frac{dw}{2x} = \int \frac{3e^w}{2x} dx.$ This simplifies $v$ further, and the substitution continues to solve it. Confirm with clearer substitution. Shall I ?