Antiderivatives with initial conditions

Problem 3.384 · hard

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