009B Sample Midterm 1, Problem 3

Evaluate the indefinite and definite integrals.

a) $\int x^{2}e^{x}dx$

b) $\int _{1}^{e}x^{3}\ln x~dx$

Foundations:
Review integration by parts

Solution:

(a)

Step 1:
We proceed using integration by parts. Let $u=x^{2}$ and $dv=e^{x}dx$. Then, $du=2xdx$ and $v=e^{x}$.
Therefore, we have
$\int x^{2}e^{x}dx=x^{2}e^{x}-\int 2xdx$

Step 2:
Now, we need to use integration by parts again. Let $u=2x$ and $dv=e^{x}dx$. Then, $du=2dx$ and $v=e^{x}$.
Therefore, we have
$\int x^{2}e^{x}dx=x^{2}e^{x}-{\bigg (}2xe^{x}-\int 2e^{x}dx{\bigg )}=x^{2}e^{x}-2xe^{x}+2e^{x}+C$

(b)

Step 1:

Step 2:

Final Answer:
(a) $x^{2}e^{x}-2xe^{x}+2e^{x}+C$
(b)

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