Lahenda matemaatika probleem

Samm-sammult

1. \int_{0}^{1} e^{- x}\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

2. u = - x,\quad du = -1\, dx

   Substitute u = - x.

3. \int e^{- x}\, dx = \int - e^{u}\, d_u

   Rewrite the integral in terms of u.

4. \int - e^{u}\, d_u = -1 \int e^{u}\, d_u

   Pull the constant -1 out of the integral.

5. \int e^{u}\, d_u = e^{u}

   ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).

6. = - e^{- x}

   Substitute back u = - x.

7. F(1) - F(0) = \left(- \frac{1}{e}\right) - \left(-1\right)

   Fundamental theorem of calculus: plug in the limits.

8. = 1 - e^{-1} \approx 0.63212

   Simplify.

Vastuse avalikustamine

1 - e^{-1}