Substitution with a given u

Problem 4.621 · hard

1. \[ \frac{d}{d x} \left(e^{2 x} + 6\right) = 2 e^{2 x} \]
   So du = 2e^(2x) dx.✓ Proved
2. In terms of u the integral becomes ∫ u**4/2 du.
   Reviewed
3. \[ \frac{d}{d u} \frac{u^{5}}{10} = \frac{u^{4}}{2} \]
   An antiderivative in u is u**5/10; differentiating confirms it.✓ Proved
4. \[ \frac{\left(e^{2 x} + 6\right)^{5}}{10} \]
   Replace u by exp(2*x) + 6.✓ Proved
5. \[ \frac{d}{d x} \frac{\left(e^{2 x} + 6\right)^{5}}{10} = \left(e^{2 x} + 6\right)^{4} e^{2 x} \]
   Differentiating the answer returns the integrand.✓ Proved