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To solve the integral \(\int x^2 e^{2x} \, dx\) numerically, we can use numerical integration techniques such as the trapezoidal rule, Simpson's rule, or Gaussian quadrature. However, since the integral is indefinite (no limits of integration are provided), we will first compute the antiderivative symbolically and then evaluate it numerically if needed. --- ### Step 1: Compute the antiderivative symbolically The integral \(\int x^2 e^{2x} \, dx\) can be solved using integration by parts. Recall the formula for integration by parts: \[ \int u \, dv = uv - \int v \, du \] Let: - \(u = x^2\), so \(du = 2x \, dx\) - \(dv = e^{2x} \, dx\), so \(v = \frac{1}{2} e^{2x}\) Applying integration by parts: \[ \int x^2 _____ _____ ____ _______ ________ ____ ____ ___ _____ _____.
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