Substitution with a given u

Problem 4.321 · hard

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