Q.The rate of new product is given by $f(x) = (100 + 2x^2)e^x$, where $x$ is the number of days the product is on the market. Find the total sale during the first four days. $(e^{-4} = 0.018)$

Integrate the daily rate from $0$ to $4$: $\int_0^4 (100+2x^2)e^{-x}dx = 104 - 152e^{-4} \approx 101.26$ units.

Note on the data. The supplied constant $e^{-4} = 0.018$ shows the rate function is $f(x) = (100 + 2x^2)e^{-x}$ (a decaying exponential); we solve accordingly, since with $e^{+x}$ the given $e^{-4}$ would be unusable. Total sale in the first four days is

$S = \int_0^4 (100 + 2x^2)\,e^{-x}\,dx.$

Step 1 — antiderivative. Using $\int e^{-x}dx = -e^{-x}$ and $\int x^2 e^{-x}dx = -(x^2+2x+2)e^{-x}$:

$\int (100 + 2x^2)e^{-x}dx = -100e^{-x} - 2(x^2+2x+2)e^{-x} = -e^{-x}(2x^2 + 4x + 104).$

Step 2 — evaluate from $0$ to $4$. …