Difference between revisions of "009B Sample Midterm 1, Problem 3"

Revision as of 14:11, 31 January 2016

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, 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 du=2xdx} 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^x~dx=x^2e^x-\int 2x~dx}

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

(b)

Step 1:
We proceed using integration by parts. Let 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 u=\ln x} 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 dv=x^3dx} . Then, 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 du=\frac{1}{x}dx} 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=\frac{x^4}{4}} .
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_{1}^{e} x^3\ln x~dx=\left.\ln x \frac{x^4}{4}\right\|_{1}^{e}-\int_1^e \frac{x^3}{4}~dx=\left.\ln x \frac{x^4}{4}-\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}}

@@ Line 1: / Line 1: @@
 <span class="exam">Evaluate the indefinite and definite integrals.
-::<span class="exam">a) <math>\int x^2 e^xdx</math>
+::<span class="exam">a) <math>\int x^2 e^x~dx</math>
 ::<span class="exam">b) <math>\int_{1}^{e} x^3\ln x~dx</math>
@@ Line 21: / Line 21: @@
 |Therefore, we have
 |-
-|<math>\int x^2 e^xdx=x^2e^x-\int 2xdx</math>
+|<math>\int x^2 e^x~dx=x^2e^x-\int 2x~dx</math>
 |}
@@ Line 31: / Line 31: @@
 |Therefore, we have
 |-
-|<math>\int x^2 e^xdx=x^2e^x-\bigg(2xe^x-\int 2e^xdx\bigg)=x^2e^x-2xe^x+2e^x+C</math>
+|<math>\int x^2 e^x~dx=x^2e^x-\bigg(2xe^x-\int 2e^x~dx\bigg)=x^2e^x-2xe^x+2e^x+C</math>
 |}
@@ Line 42: / Line 42: @@
 |Therefore, we have
 |-
-|<math>\int_{1}^{e} x^3\ln x~dx=\left.\ln x \frac{x^4}{4}\right|_{1}^{e}-\int_1^e \frac{x^3}{4}dx=\left.\ln x \frac{x^4}{4}-\frac{x^4}{16}\right|_{1}^{e}</math>
+|<math>\int_{1}^{e} x^3\ln x~dx=\left.\ln x \frac{x^4}{4}\right|_{1}^{e}-\int_1^e \frac{x^3}{4}~dx=\left.\ln x \frac{x^4}{4}-\frac{x^4}{16}\right|_{1}^{e}</math>
 |-
 |