Substitution with a given u

Problem 4.482 · hard

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