(a) ∫ 𝑥2𝑒−𝑥𝑑𝑥

To solve the integral ∫ 𝑥²𝑒⁻ˣ𝑑𝑥, we can use the method of integration by parts. The formula for integration by parts is ∫udv = uv - ∫vdu.

Step 1: Choose u and dv Let's choose u = x² and dv = e⁻ˣdx.

Step 2: Compute du and v Differentiating u, we get du = 2xdx. Integrating dv, we get v = -e⁻ˣ.

Step 3: Apply the integration by parts formula ∫udv = uv - ∫vdu becomes ∫x²e⁻ˣdx = -x²e⁻ˣ - ∫(-e⁻ˣ * 2x)dx.

Step 4: Simplify the integral The integral becomes -x²e⁻ˣ + 2∫xe⁻ˣdx.

Step 5: Apply integration by parts again For the integral ∫xe⁻ˣdx, let's choose u = x and dv = e⁻ˣdx. Then du = dx and v = -e⁻ˣ. Applying the integration by parts formula, we get ∫xe⁻ˣdx = -xe⁻ˣ - ∫(-e⁻ˣ)dx = -xe⁻ˣ + ∫e⁻ˣdx.

Step 6: Simplify the integral The integral becomes -x²e⁻ˣ + 2(-xe⁻ˣ + ∫e⁻ˣdx) = -x²e⁻ˣ - 2xe⁻ˣ + 2∫e⁻ˣdx.

Step 7: Final integration The integral ∫e⁻ˣdx is -e⁻ˣ. Substituting this in, we get -x²e⁻ˣ - 2xe⁻ˣ + 2(-e⁻ˣ) = -x²e⁻ˣ - 2xe⁻ˣ - 2e⁻ˣ.

So, ∫ 𝑥²𝑒⁻ˣ𝑑𝑥 = -x²e⁻ˣ - 2xe⁻ˣ - 2e⁻ˣ + C, where C is the constant of integration.