Antiderivatives with initial conditions

Problem 3.475 · hard

1. \[ \frac{d}{d x} \left(x^{4} - 3 e^{x}\right) = 4 x^{3} - 3 e^{x} \]
   An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
2. \[ - \left. x^{4} - 3 e^{x} \right|_{\substack{ x=0 }} - 6 = -3 \]
   f′(0) = -6 fixes C₁.✓ Proved
3. \[ \frac{d}{d x} \left(\frac{x^{5}}{5} - 3 x - 3 e^{x}\right) = x^{4} - 3 e^{x} - 3 \]
   Antidifferentiate f′; add a constant C₀.✓ Proved
4. \[ - \left. \frac{x^{5}}{5} - 3 x - 3 e^{x} \right|_{\substack{ x=0 }} - 4 = -1 \]
   f(0) = -4 fixes C₀.✓ Proved
5. \[ \frac{x^{5}}{5} - 3 x - 3 e^{x} - 1 \]
   The solution.✓ Proved