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
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 2xdx}

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

(b)

Step 1:

Step 2:

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)

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