Evaluate the following integral using integration by parts. \[ \int x^{2} e^{12 x} d x \]

To evaluate the integral \(\int x^{2} e^{12 x} d x\) using integration by parts, we apply the integration by parts formula, which is given by: \[ \int u \, dv = uv - \int v \, du \] We start by choosing: - \(u = x^2\) which means \(du = 2x \, dx\) - \(dv = e^{12x} \, dx\) which gives us \(v = \frac{1}{12} e^{12x}\) Now we can apply the integration by parts formula: \[ \int x^{2} e^{12x} \, dx = uv - \int v \, du \] Substituting the values we found: \[ = x^2 \cdot \frac{1}{12} e^{12x} - \int \frac{1}{12} e^{12x} \cdot 2x \, dx \] This simplifies to: \[ = \frac{x^2}{12} e^{12x} - \frac{1}{6} \int x e^{12x} \, dx \] Now, we need to solve \(\int x e^{12x} \, dx\) using integration by parts again. Let’s choose: - \(u = x\) which means \(du = dx\) - \(dv = e^{12x} \, dx\) which means \(v = \frac{1}{12} e^{12x}\) Applying the integration by parts formula again: \[ \int x e^{12x} \, dx = uv - \int v \, du \] Substituting the values: \[ = x \cdot \frac{1}{12} e^{12x} - \int \frac{1}{12} e^{12x} \cdot 1 \, dx \] This simplifies to: \[ = \frac{x}{12} e^{12x} - \frac{1}{12} \int e^{12x} \, dx \] The integral of \(e^{12x}\) is: \[ \int e^{12x} \, dx = \frac{1}{12} e^{12x} + C \] So we have: \[ \int x e^{12x} \, dx = \frac{x}{12} e^{12x} - \frac{1}{144} e^{12x} + C \] Now substituting back into our previous expression for \(\int x^2 e^{12x} \, dx\): \[ \int x^2 e^{12x} \, dx = \frac{x^2}{12} e^{12x} - \frac{1}{6} \left( \frac{x}{12} e^{12x} - \frac{1}{144} e^{12x} \right) \] This simplifies to: \[ = \frac{x^2}{12} e^{12x} - \frac{1}{72} x e^{12x} + \frac{1}{864} e^{12x} + C \] Combining everything, we have the final result: \[ \int x^{2} e^{12x} \, dx = \frac{e^{12x}}{864} (72x^2 - 12x + 1) + C \]