Antiderivatives with initial conditions

Problem 3.484 · hard

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