Chapter 6.3, Problem 6E

To determine

To find: The value of indefinite integral.

Answer to Problem 6E

The indefinite integral: $-e^{-x}\left(x^2+2x+2\right)+C$

Explanation of Solution

Given information:

The integral is: $\int x^2{e}^{-x}dx$

Calculation:

The given integral is: $\int x^2{e}^{-x}dx$

Repeated integration by parts will be required since the second function, $e^{-x}$ , is cyclical and the first function, $x^2$ , is a power of $x$ higher than 1. To simplify the calculations, use tabular integration by letting

  $\begin{array}{c}f(x)=x^2\\ g(x)=e^{-x}\end{array}$

Then find the derivatives of $x^2$ until reach 0 and the anti derivatives of $e^{-x}$

Then, draw arrows connecting each first-column expression (except from the final expression of 0) to the expression in the second-column row below. Start with a positive sign and alternate between positive and negative signs as write the signs above the arrows.

  

Utilize the operation signals located above each arrow to combine the results of the functions that are connected by arrows. Remember to add the $+C$ .

  $\begin{array}{c}\int x^2{e}^{-x}dx=+x^2\left(-e^{-x}\right)-2x{e}^{-x}+2\left(-e^{-x}\right)+C\\ =-x^2{e}^{-x}-2x{e}^{-x}-2e^{-x}+C\\ =-e^{-x}\left(x^2+2x+2\right)+C\end{array}$

Therefore the required indefinite integral is $-e^{-x}\left(x^2+2x+2\right)+C$ .