Antiderivatives with initial conditions

Problem 3.481 · hard

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