Substitution with a given u

Problem 4.941 · hard

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