Antiderivatives with initial conditions

Problem 3.432 · hard

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