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:
Integration by parts tells us that $\int u~dv=uv-\int v~du.$
How would you integrate $\int x\ln x~dx?$
You could use integration by parts.
Let $u=\ln x$ and $dv=x~dx.$ Then, $du={\frac {1}{x}}dx$ and $v={\frac {x^{2}}{2}}.$
Thus, $\int x\ln x~dx\,=\,{\frac {x^{2}\ln x}{2}}-\int {\frac {x}{2}}~dx\,=\,{\frac {x^{2}\ln x}{2}}-{\frac {x^{2}}{4}}+C.$

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
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int x^{2}e^{x}~dx=x^{2}e^{x}-\int 2xe^{x}~dx} .

Step 2:
Now, we need to use integration by parts again. Let $u=2x$ and $dv=e^{x}dx$. Then, $du=2dx$ and $v=e^{x}$.
Building on the previous step, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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:
We proceed using integration by parts. Let $u=\ln x$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dv=x^{3}dx} . Then, $du={\frac {1}{x}}dx$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v={\frac {x^{4}}{4}}} .
Therefore, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{1}^{e}x^{3}\ln x~dx=\left.\ln x{\bigg (}{\frac {x^{4}}{4}}{\bigg )}\right|_{1}^{e}-\int _{1}^{e}{\frac {x^{3}}{4}}~dx=\left.\ln x{\bigg (}{\frac {x^{4}}{4}}{\bigg )}-{\frac {x^{4}}{16}}\right|_{1}^{e}} .

Step 2:
Now, we evaluate to get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{1}^{e}x^{3}\ln x~dx={\bigg (}(\ln e){\frac {e^{4}}{4}}-{\frac {e^{4}}{16}}{\bigg )}-{\bigg (}(\ln 1){\frac {1^{4}}{4}}-{\frac {1^{4}}{16}}{\bigg )}={\frac {e^{4}}{4}}-{\frac {e^{4}}{16}}+{\frac {1}{16}}={\frac {3e^{4}+1}{16}}} .

Final Answer:
(a)   $x^{2}e^{x}-2xe^{x}+2e^{x}+C$
(b)   ${\frac {3e^{4}+1}{16}}$

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