Substitution with a given u

Problem 4.481 · hard

1. \[ \frac{d}{d x} \left(x^{3} + 4\right) = 3 x^{2} \]
   So du = 3x² dx.✓ Proved
2. In terms of u the integral becomes ∫ 4*u**2/3 du.
   Reviewed
3. \[ \frac{d}{d u} \frac{4 u^{3}}{9} = \frac{4 u^{2}}{3} \]
   An antiderivative in u is 4*u**3/9; differentiating confirms it.✓ Proved
4. \[ \frac{4 \left(x^{3} + 4\right)^{3}}{9} \]
   Replace u by x**3 + 4.✓ Proved
5. \[ \frac{d}{d x} \frac{4 \left(x^{3} + 4\right)^{3}}{9} = 4 x^{2} \left(x^{3} + 4\right)^{2} \]
   Differentiating the answer returns the integrand.✓ Proved