Antiderivatives with initial conditions

Problem 3.431 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{4}}{4} - \frac{3}{x}\right) = - x^{3} + \frac{3}{x^{2}} \]
   An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
2. \[ 1 - \left. \frac{\left(-1\right) x^{4}}{4} - \frac{3}{x} \right|_{\substack{ x=1 }} = \frac{17}{4} \]
   f′(1) = 1 fixes C₁.✓ Proved
3. \[ \frac{d}{d x} \left(- \frac{x^{5}}{20} + \frac{17 x}{4} - 3 \ln{\left(x \right)}\right) = - \frac{x^{4}}{4} + \frac{17}{4} - \frac{3}{x} \]
   Antidifferentiate f′; add a constant C₀.✓ Proved
4. \[ 3 - \left. \frac{17 x}{4} + \frac{\left(-1\right) x^{5}}{20} - 3 \ln{\left(x \right)} \right|_{\substack{ x=1 }} = - \frac{6}{5} \]
   f(1) = 3 fixes C₀.✓ Proved
5. \[ \frac{17 x}{4} + \frac{\left(-1\right) x^{5}}{20} - 3 \ln{\left(x \right)} - \frac{6}{5} = - \frac{x^{5}}{20} + \frac{17 x}{4} - 3 \ln{\left(x \right)} - \frac{6}{5} \]
   The solution.✓ Proved