## MATH1003 Quizzes

Quiz 4: Integration by Parts
Question 1 Questions
Recall that integration by parts is a technique to re-express the integral of a product of two functions $u$ and $\frac{dv}{dx}$ in a form which allows it to be more easily evaluated. The formula is $\int \phantom{\rule{1em}{0ex}}u\phantom{\rule{2.77695pt}{0ex}}\frac{dv}{dx}\phantom{\rule{2.77695pt}{0ex}}dx=uv-\int \phantom{\rule{1em}{0ex}}v\phantom{\rule{2.77695pt}{0ex}}\frac{du}{dx}\phantom{\rule{2.77695pt}{0ex}}dx$. When applying the method of integration by parts to find $\int \phantom{\rule{1em}{0ex}}x{e}^{x}\phantom{\rule{1em}{0ex}}dx$, the best choice of $u$ and $\frac{dv}{dx}$ is Exactly one option must be correct)
 a) $u={e}^{x},\phantom{\rule{1em}{0ex}}\frac{dv}{dx}=x$ b) $u=x{e}^{x},\phantom{\rule{1em}{0ex}}\frac{dv}{dx}=1$ c) $u=1,\phantom{\rule{1em}{0ex}}\frac{dv}{dx}=x{e}^{x}$ d) $u=x,\phantom{\rule{1em}{0ex}}\frac{dv}{dx}={e}^{x}$ e) None of the above

Choice (a) is incorrect
This expresses the original integral in terms of a new integral which is even harder! After applying integration by parts as suggested, we have $\int \phantom{\rule{1em}{0ex}}x{e}^{x}\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{x}^{2}{e}^{x}-\int \phantom{\rule{2.77695pt}{0ex}}\frac{1}{2}{x}^{2}{e}^{x}\phantom{\rule{2.77695pt}{0ex}}dx$.
Choice (b) is incorrect
This expresses the original integral in terms of a new integral which is even harder! After applying integration by parts as suggested, we have $\int \phantom{\rule{1em}{0ex}}x{e}^{x}\phantom{\rule{1em}{0ex}}dx={x}^{2}{e}^{x}-\int \phantom{\rule{2.77695pt}{0ex}}\left({x}^{2}{e}^{x}+x{e}^{x}\right)\phantom{\rule{2.77695pt}{0ex}}dx$.
Choice (c) is incorrect
The problem of finding $v$ if $dv=x{e}^{x}\phantom{\rule{2.77695pt}{0ex}}dx$ is exactly the integral we started with! So no progress in this case.
Choice (d) is correct!
This gives genuine simplification and results in an easy integral. We obtain $\int \phantom{\rule{2.77695pt}{0ex}}x{e}^{x}\phantom{\rule{2.77695pt}{0ex}}dx=x{e}^{x}-\int \phantom{\rule{2.77695pt}{0ex}}{e}^{x}\phantom{\rule{2.77695pt}{0ex}}dx$.
Choice (e) is incorrect
Find ${\int }_{0}^{1}\phantom{\rule{1em}{0ex}}x{e}^{x}\phantom{\rule{1em}{0ex}}dx$ using integration by parts, and enter your answer.

Correct!
Choosing $u=x,\phantom{\rule{1em}{0ex}}\frac{dv}{dx}={e}^{x}$ gives ${\int }_{0}^{1}\phantom{\rule{1em}{0ex}}x{e}^{x}\phantom{\rule{1em}{0ex}}dx={\left[x{e}^{x}\right]}_{0}^{1}-{\int }_{0}^{1}\phantom{\rule{2.77695pt}{0ex}}{e}^{x}\phantom{\rule{2.77695pt}{0ex}}dx={\left[{e}^{x}\left(x-1\right)\right]}_{0}^{1}=1.$
Try choosing $u=x,\phantom{\rule{1em}{0ex}}\frac{dv}{dx}={e}^{x}$.
The reduction formula for ${I}_{n}=\int \phantom{\rule{2.77695pt}{0ex}}x{\left(lnx\right)}^{n}\phantom{\rule{2.77695pt}{0ex}}dx$ is ${I}_{n}=\frac{1}{2}{x}^{2}{\left(lnx\right)}^{n}-\frac{n}{2}{I}_{n-1}$. Given that ${I}_{0}=\frac{1}{2}{x}^{2}+C$, find ${\int }_{1}^{e}\phantom{\rule{2.77695pt}{0ex}}x\left(lnx\right)\phantom{\rule{2.77695pt}{0ex}}dx$. Give your answer correct to three decimal places.

Correct!
The integral ${I}_{1}$ is $\frac{1}{2}{x}^{2}lnx-\frac{1}{4}{x}^{2}+C$, using the reduction formula with $n=1$. Evaluating this between $1$ and $e$ gives 2.097 to three decimal places.
Substitute $n=1$ into the reduction formula to obtain ${I}_{1}$.
Which option is an antiderivative of ${x}^{4}{e}^{x}$ ? Use the reduction formula ${I}_{n}=\int \phantom{\rule{2.77695pt}{0ex}}{x}^{n}{e}^{x}\phantom{\rule{2.77695pt}{0ex}}dx={x}^{n}{e}^{x}-n{I}_{n-1}$ to help answer this question. Exactly one option must be correct)
 a) ${e}^{x}{x}^{4}+4{x}^{3}{e}^{x}-12{x}^{2}{e}^{x}+24x{e}^{x}-24{e}^{x}$ b) ${e}^{x}\left({x}^{4}-4{x}^{3}-12{x}^{2}-24x+24\right)$ c) ${e}^{x}{x}^{4}-4{x}^{3}{e}^{x}+12{x}^{2}{e}^{x}-24x{e}^{x}+24{e}^{x}+C{e}^{x}$ d) ${e}^{x}\left({x}^{4}-4{x}^{3}+12{x}^{2}-24x+24\right)+15$ e) None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Which option equals ${\int }_{0}^{1}\phantom{\rule{2.77695pt}{0ex}}x{tan}^{-1}x\phantom{\rule{2.77695pt}{0ex}}dx$ ? (Hint: use integration by parts with $u={tan}^{-1}x$ and $\frac{dv}{dx}=x$.) Exactly one option must be correct)
 a) $\frac{\pi }{4}-\frac{1}{2}$ b) $-\frac{1}{2}$ c) $\frac{\pi }{8}-\frac{1}{2}$ d) $\frac{1}{2}\left(\frac{\pi }{4}+1\right)$ e) $\frac{\pi }{4}$

Choice (a) is correct!
The indefinite integral is $\frac{1}{2}{x}^{2}{tan}^{-1}x-\frac{1}{2}x+\frac{1}{2}{tan}^{-1}x+C$, which, when evaluated between 0 and 1, matches this option.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which option equals ${\int }_{0}^{\frac{1}{2}}\phantom{\rule{2.77695pt}{0ex}}{sin}^{-1}x\phantom{\rule{2.77695pt}{0ex}}dx$ ? (Hint: use integration by parts with $u={sin}^{-1}x$ and $\frac{dv}{dx}=1$.) Exactly one option must be correct)
 a) $\frac{\pi }{12}-\frac{\sqrt{3}}{2}+1$ b) $\frac{\pi }{4}+\frac{\sqrt{3}}{2}$ c) $\frac{\pi }{12}+\frac{\sqrt{3}}{2}-1$ d) $\frac{\pi }{3}-1$ e) $\frac{\pi }{12}-\frac{\sqrt{3}}{2}-1$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The indefinite integral is $x{sin}^{-1}x+\sqrt{1-{x}^{2}}+C$, which, when evaluated between $0$ and $\frac{1}{2}$, matches this option.
Choice (d) is incorrect
Choice (e) is incorrect
The finite area bounded by the curve $y=lnx$, the line $y=1$ and the tangent line to $y=lnx$ at $x=1$ is given as an integral with respect to $x$ by ${\int }_{1}^{2}\phantom{\rule{2.77695pt}{0ex}}\left(x-1-lnx\right)\phantom{\rule{2.77695pt}{0ex}}dx+{\int }_{2}^{e}\phantom{\rule{2.77695pt}{0ex}}\left(1-lnx\right)\phantom{\rule{2.77695pt}{0ex}}dx$. Which option equals the same area given as an integral with respect to $y$ ? (You must draw a sketch to help you with this question.) Exactly one option must be correct)
 a) ${\int }_{1}^{e}\phantom{\rule{2.77695pt}{0ex}}{e}^{y}-\left(y+1\right)\phantom{\rule{2.77695pt}{0ex}}dy$ b) ${\int }_{0}^{1}\phantom{\rule{2.77695pt}{0ex}}lny-y+1\phantom{\rule{2.77695pt}{0ex}}dy$ c) ${\int }_{0}^{e}\phantom{\rule{2.77695pt}{0ex}}{e}^{y}-y+1\phantom{\rule{2.77695pt}{0ex}}dy$ d) ${\int }_{0}^{1}\phantom{\rule{2.77695pt}{0ex}}{e}^{y}-\left(y+1\right)\phantom{\rule{2.77695pt}{0ex}}dy$ e) None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Which option equals $\int \phantom{\rule{2.77695pt}{0ex}}x{sec}^{2}x\phantom{\rule{2.77695pt}{0ex}}dx$ ? Exactly one option must be correct)
 a) $xsecxtanx-ln\left(cosx\right)+C$ b) $xtanx+ln|cosx|+C$ c) $x{tan}^{2}x-ln|cosx|+C$ d) $xtanx-ln\left(cosx\right)+C$ e) None of the above

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
In some problems you need to apply the integration by parts method twice in order to obtain the required answer. The integral $\int \phantom{\rule{2.77695pt}{0ex}}sin\left(lnx\right)\phantom{\rule{2.77695pt}{0ex}}dx$ is one such problem. Which of the following options gives the expression obtained after one application of integration by parts? Exactly one option must be correct)
 a) $xsin\phantom{\rule{0.3em}{0ex}}\left(lnx\right)-\int \phantom{\rule{2.77695pt}{0ex}}cos\phantom{\rule{0.3em}{0ex}}\left(lnx\right)\phantom{\rule{2.77695pt}{0ex}}dx$ b) $-cosxlnx+\int \phantom{\rule{2.77695pt}{0ex}}\frac{cosx}{x}\phantom{\rule{2.77695pt}{0ex}}dx$ c) $-xsin\phantom{\rule{0.3em}{0ex}}\left(lnx\right)+\int \phantom{\rule{2.77695pt}{0ex}}xcos\phantom{\rule{0.3em}{0ex}}\left(lnx\right)\phantom{\rule{2.77695pt}{0ex}}dx$ d) $xsin\phantom{\rule{0.3em}{0ex}}\left(lnx\right)-\int \phantom{\rule{2.77695pt}{0ex}}\frac{sin\phantom{\rule{0.3em}{0ex}}\left(lnx\right)}{x}\phantom{\rule{2.77695pt}{0ex}}dx$ e) $cosxlnx+\int \phantom{\rule{2.77695pt}{0ex}}\frac{cosx}{x}\phantom{\rule{2.77695pt}{0ex}}dx$

Choice (a) is correct!
Choice (b) is incorrect
Try $u=sin\phantom{\rule{0.3em}{0ex}}\left(lnx\right)$ and $\frac{dv}{dx}=1$ in the integration by parts formula.
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which option equals ${\int }_{1}^{e}\phantom{\rule{2.77695pt}{0ex}}sin\left(lnx\right)\phantom{\rule{2.77695pt}{0ex}}dx$  ? Exactly one option must be correct)
 a) $\frac{1}{2}\left(esin1+ecos1\right)$ b) $sin1-cos1$ c) $\frac{1}{2}\left(esin1-ecos1+1\right)$ d) $\frac{1}{2}sin1-2ecos1+1$ e) $esin1-ecos1$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choice (e) is incorrect