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
$\int x^{2}e^{x}~dx=x^{2}e^{x}-\int 2xe^{x}~dx$.

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
$\int x^{2}e^{x}~dx=x^{2}e^{x}-{\bigg (}2xe^{x}-\int 2e^{x}~dx{\bigg )}=x^{2}e^{x}-2xe^{x}+2e^{x}+C$.

(b)

Step 1:
We proceed using integration by parts. Let $u=\ln x$ and $dv=x^{3}dx$. Then, $du={\frac {1}{x}}dx$ and $v={\frac {x^{4}}{4}}$.
Therefore, we have
$\int _{1}^{e}x^{3}\ln x~dx=\left.\ln x{\bigg (}{\frac {x^{4}}{4}}{\bigg )}\right|_{1}^{e}-\int _{1}^{e}{\frac {x^{3}}{4}}~dx=\left.\ln x{\bigg (}{\frac {x^{4}}{4}}{\bigg )}-{\frac {x^{4}}{16}}\right|_{1}^{e}$.

Step 2:
Now, we evaluate to get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2e^x-2xe^x+2e^x+C}
(b)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3e^4+1}{16}}

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