Substitution with a given u

Problem 4.483 · hard

1. \[ \frac{d}{d x} \left(x + 5\right) = 1 \]
   So du = dx, and x = u − 5.✓ Proved
2. In terms of u the integral becomes ∫ sqrt(u)*(6*u - 30) du.
   Reviewed
3. \[ \frac{d}{d u} \left(\frac{12 u^{\frac{5}{2}}}{5} - 20 u^{\frac{3}{2}}\right) = \sqrt{u} \left(6 u - 30\right) \]
   An antiderivative in u is 12*u**(5/2)/5 - 20*u**(3/2); differentiating confirms it.✓ Proved
4. \[ \frac{12 \left(x + 5\right)^{\frac{5}{2}}}{5} - 20 \left(x + 5\right)^{\frac{3}{2}} \]
   Replace u by x + 5.✓ Proved
5. \[ \frac{d}{d x} \left(\frac{12 \left(x + 5\right)^{\frac{5}{2}}}{5} - 20 \left(x + 5\right)^{\frac{3}{2}}\right) = 6 x \sqrt{x + 5} \]
   Differentiating the answer returns the integrand.✓ Proved