Antiderivatives with initial conditions

Problem 3.376 · hard

1. \[ \frac{d}{d x} \left(\frac{x^{4}}{4} + 4 \cos{\left(x \right)}\right) = x^{3} - 4 \sin{\left(x \right)} \]
   An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
2. \[ 6 - \left. \frac{x^{4}}{4} + 4 \cos{\left(x \right)} \right|_{\substack{ x=1 }} = \frac{23}{4} - 4 \cos{\left(1 \right)} \]
   f′(1) = 6 fixes C₁.✓ Proved
3. \[ \frac{d}{d x} \left(\frac{x^{5}}{20} - 4 x \cos{\left(1 \right)} + \frac{23 x}{4} + 4 \sin{\left(x \right)}\right) = \frac{x^{4}}{4} + 4 \cos{\left(x \right)} - 4 \cos{\left(1 \right)} + \frac{23}{4} \]
   Antidifferentiate f′; add a constant C₀.✓ Proved
4. \[ 1 - \left. \frac{x^{5}}{20} - 4 x \cos{\left(1 \right)} + \frac{23 x}{4} + 4 \sin{\left(x \right)} \right|_{\substack{ x=1 }} = - \frac{24}{5} - 4 \sin{\left(1 \right)} + 4 \cos{\left(1 \right)} \]
   f(1) = 1 fixes C₀.✓ Proved
5. \[ \frac{x^{5}}{20} - 4 x \cos{\left(1 \right)} + \frac{23 x}{4} + 4 \sin{\left(x \right)} - \frac{24}{5} - 4 \sin{\left(1 \right)} + 4 \cos{\left(1 \right)} \]
   The solution.✓ Proved