Substitution with a given u

Problem 4.777 · hard

1. \[ \frac{d}{d x} \left(e^{2 x} + 3\right) = 2 e^{2 x} \]
   So du = 2e^(2x) dx.✓ Proved
2. In terms of u the integral becomes ∫ 3*u**3/2 du.
   Reviewed
3. \[ \frac{d}{d u} \frac{3 u^{4}}{8} = \frac{3 u^{3}}{2} \]
   An antiderivative in u is 3*u**4/8; differentiating confirms it.✓ Proved
4. \[ \frac{3 \left(e^{2 x} + 3\right)^{4}}{8} \]
   Replace u by exp(2*x) + 3.✓ Proved
5. \[ \frac{d}{d x} \frac{3 \left(e^{2 x} + 3\right)^{4}}{8} = 3 \left(e^{2 x} + 3\right)^{3} e^{2 x} \]
   Differentiating the answer returns the integrand.✓ Proved