Evaluate the integral.
- (Express your answer in terms for e.)

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts. Letting $u=x^{-x}$ and $dv=e^{x}dx$, we get $du/dx=(-1-x)x^{-x-1}$ and $v=e^{x}$. Applying the integration by parts formula, we get:

$$\int x^{-x}e^{x}dx=(-x^{-x}e^{x})-\int(-1-x)x^{-x-1}e^{x}dx$$

Simplifying, we get:

$$\int x^{-x}e^{x}dx=x^{-x}(-e^{x}+\int(1+x)x^{-x-1}e^{x}dx)$$

Letting $u=x^{-x-1}$ and $dv=e^{x}(1+x)dx$, we get $du/dx=(-x^{-x-2}-x^{-x-1}\ln x)$ and $v=e^{x}(x+2)$. Applying the integration by parts formula again, we get:

$$\int (1+x)x^{-x-1}e^{x}dx=(x^{-x}e^{x}(x+2))-\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

Simplifying, we get:

$$\int (1+x)x^{-x-1}e^{x}dx=x^{-x}e^{x}(x+2)-\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

Substituting this into our original integral, we get:

$$\int x^{-x}e^{x}dx=x^{-x}(-e^{x}+x+2)-x^{-x}e^{x}(x+2)+\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

Simplifying, we get:

$$\int x^{-x}e^{x}dx=-x^{-x}e^{x}+\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

This integral cannot be simplified further, so our answer is $\boxed{	extbf{(D) }11ebb6d7_{48aa_aa42_b437_695d262f3613}}$.

Evaluate the Integral. - (Express Your Answer in Terms for E.)