Antiderivatives with initial conditions

Problem 3.529 · hard

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