$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 2.1E: Exercises

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Integration by Parts

In using the technique of integration by parts, you must carefully choose which expression is $$u$$. For each of the following problems, use the guidelines in this section to choose $$u$$. Do not evaluate the integrals.

### Exercise $$\PageIndex{1}$$

$$\displaystyle ∫x^3e^{2x}\,dx$$

$$u=x^3$$

### Exercise $$\PageIndex{2}$$

$$\displaystyle ∫x^3\ln(x)\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{3}$$

$$\displaystyle ∫y^3\cos y\,dy$$

$$u=y^3$$

### Exercise $$\PageIndex{4}$$

$$\displaystyle ∫x^2\arctan x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{5}$$

$$\displaystyle ∫e^{3x}\sin(2x)\,dx$$

$$u=\sin(2x)$$

In exercises 6 - 37, find the integral by using the simplest method. Not all problems require integration by parts.

### Exercise $$\PageIndex{6}$$

$$\displaystyle ∫v\sin v\,dv$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{7}$$

(\displaystyle ∫\ln x\,dx\)

(Hint: $$\displaystyle ∫\ln x\,dx$$ is equivalent to $$\displaystyle ∫1⋅\ln(x)\,dx.)$$

$$\displaystyle ∫\ln x\,dx \quad = \quad−x+x\ln x+C$$

### Exercise $$\PageIndex{8}$$

$$\displaystyle ∫x\cos x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{9}$$

$$\displaystyle ∫\tan^{−1}x\,dx$$

$$\displaystyle ∫\tan^{−1}x\,dx\quad = \quad x\tan^{−1}x−\tfrac{1}{2}\ln(1+x^2)+C$$

### Exercise $$\PageIndex{10}$$

$$\displaystyle ∫x^2e^x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{11}$$

$$\displaystyle ∫x\sin(2x)\,dx$$

$$\displaystyle ∫x\sin(2x)\,dx \quad = \quad −\tfrac{1}{2}x\cos(2x)+\tfrac{1}{4}\sin(2x)+C$$

### Exercise $$\PageIndex{12}$$

$$\displaystyle ∫xe^{4x}\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{13}$$

$$\displaystyle ∫xe^{−x}\,dx$$

$$\displaystyle ∫xe^{−x}\,dx \quad = \quad e^{−x}(−1−x)+C$$

### Exercise $$\PageIndex{14}$$

$$\displaystyle ∫x\cos 3x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{15}$$

$$\displaystyle ∫x^2\cos x\,dx$$

$$\displaystyle ∫x^2\cos x\,dx \quad = \quad 2x\cos x+(−2+x^2)\sin x+C$$

### Exercise $$\PageIndex{16}$$

$$\displaystyle ∫x\ln x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{17}$$

$$\displaystyle ∫\ln(2x+1)\,dx$$

$$\displaystyle ∫\ln(2x+1)\,dx \quad = \quad \tfrac{1}{2}(1+2x)(−1+\ln(1+2x))+C$$

### Exercise $$\PageIndex{18}$$

$$\displaystyle ∫x^2e^{4x}\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{19}$$

$$\displaystyle ∫e^x\sin x\,dx$$

$$\displaystyle ∫e^x\sin x\,dx \quad = \quad \tfrac{1}{2}e^x(−\cos x+\sin x)+C$$

### Exercise $$\PageIndex{20}$$

$$\displaystyle ∫e^x\cos x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{21}$$

$$\displaystyle ∫xe^{−x^2}\,dx$$

$$\displaystyle ∫xe^{−x^2}\,dx \quad = \quad −\frac{e^{−x^2}}{2}+C$$

### Exercise $$\PageIndex{22}$$

$$\displaystyle ∫x^2e^{−x}\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{23}$$

$$\displaystyle ∫\sin(\ln(2x))\,dx$$

$$\displaystyle ∫\sin(\ln(2x))\,dx \quad = \quad −\tfrac{1}{2}x\cos[\ln(2x)]+\tfrac{1}{2}x\sin[\ln(2x)]+C$$

### Exercise $$\PageIndex{24}$$

$$\displaystyle ∫\cos(\ln x)\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{25}$$

$$\displaystyle ∫(\ln x)^2\,dx$$

$$\displaystyle ∫(\ln x)^2\,dx \quad = \quad 2x−2x\ln x+x(\ln x)^2+C$$

### Exercise $$\PageIndex{26}$$

$$\displaystyle ∫\ln(x^2)\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{27}$$

$$\displaystyle ∫x^2\ln x\,dx$$

$$\displaystyle ∫x^2\ln x\,dx \quad = \quad −\frac{x^3}{9}+\tfrac{1}{3}x^3\ln x+C$$

### Exercise $$\PageIndex{28}$$

$$\displaystyle ∫\sin^{−1}x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{29}$$

$$\displaystyle ∫\cos^{−1}(2x)\,dx$$

$$\displaystyle ∫\cos^{−1}(2x)\,dx \quad = \quad −\tfrac{1}{2}\sqrt{1−4x^2}+x\cos^{−1}(2x)+C$$

### Exercise $$\PageIndex{30}$$

$$\displaystyle ∫x\arctan x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{31}$$

$$\displaystyle ∫x^2\sin x\,dx$$

$$\displaystyle ∫x^2\sin x\,dx \quad = \quad −(−2+x^2)\cos x+2x\sin x+C$$

### Exercise $$\PageIndex{32}$$

$$\displaystyle ∫x^3\cos x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{33}$$

$$\displaystyle ∫x^3\sin x\,dx$$

$$\displaystyle ∫x^3\sin x\,dx \quad = \quad −x(−6+x^2)\cos x+3(−2+x^2)\sin x+C$$

### Exercise $$\PageIndex{34}$$

$$\displaystyle ∫x^3e^x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{35}$$

$$\displaystyle ∫x\sec^{−1}x\,dx$$

$$\displaystyle ∫x\sec^{−1}x\,dx \quad = \quad \tfrac{1}{2}x\left(−\sqrt{1−\frac{1}{x^2}}+x⋅\sec^{−1}x\right)+C$$

### Exercise $$\PageIndex{36}$$

$$\displaystyle ∫x\sec^2x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{37}$$

$$\displaystyle ∫x\cosh x\,dx$$

$$\displaystyle ∫x\cosh x\,dx \quad = \quad −\cosh x+x\sinh x+C$$

In exercises 38 - 46, compute the definite integrals. Use a graphing utility to confirm your answers.

### Exercise $$\PageIndex{38}$$

$$\displaystyle ∫^1_{1/e}\ln x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{39}$$

$$\displaystyle ∫^1_0xe^{−2x}\,dx$$ (Express the answer in exact form.)

$$\displaystyle ∫^1_0xe^{−2x}\,dx \quad = \quad \frac{1}{4}−\frac{3}{4e^2}$$

### Exercise $$\PageIndex{40}$$

$$\displaystyle ∫^1_0e^{\sqrt{x}}\,dx \quad (\text{let}\, u=\sqrt{x})$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{41}$$

$$\displaystyle ∫^e_1\ln(x^2)\,dx$$

$$\displaystyle ∫^e_1\ln(x^2)\,dx \quad = \quad 2$$

### Exercise $$\PageIndex{42}$$

$$\displaystyle ∫^π_0x\cos x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{43}$$

$$\displaystyle ∫^π_{−π}x\sin x\,dx$$ (Express the answer in exact form.)

$$\displaystyle ∫^π_{−π}x\sin x\,dx \quad = \quad 2\pi$$

### Exercise $$\PageIndex{44}$$

$$\displaystyle ∫^3_0\ln(x^2+1)\,dx$$ (Express the answer in exact form.)

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{45}$$

$$\displaystyle ∫^{π/2}_0x^2\sin x\,dx$$ (Express the answer in exact form.)

$$\displaystyle ∫^{π/2}_0x^2\sin x\,dx \quad = \quad −2+π$$

### Exercise $$\PageIndex{46}$$

$$\displaystyle ∫^1_0x5^x\,dx$$ (Express the answer using five significant digits.)

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{47}$$

Evaluate $$\displaystyle ∫\cos x\ln(\sin x)\,dx$$

$$\displaystyle ∫\cos x\ln(\sin x)\,dx \quad = \quad −\sin(x)+\ln[\sin(x)]\sin x+C$$

In exercises 48 - 50, derive the following formulas using the technique of integration by parts. Assume that $$n$$ is a positive integer. These formulas are called reduction formulas because the exponent in the $$x$$ term has been reduced by one in each case. The second integral is simpler than the original integral.

### Exercise $$\PageIndex{48}$$

$$\displaystyle ∫x^ne^x\,dx=x^ne^x−n∫x^{n−1}e^x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{49}$$

$$\displaystyle ∫x^n\cos x\,dx=x^n\sin x−n∫x^{n−1}\sin x\,dx$$

### Exercise $$\PageIndex{50}$$

$$\displaystyle ∫x^n\sin x\,dx=$$______

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{51}$$

Integrate $$\displaystyle ∫2x\sqrt{2x−3}\,dx$$ using two methods:

a. Using parts, letting $$dv=\sqrt{2x−3}\,dx$$

b. Substitution, letting $$u=2x−3$$

a. $$\displaystyle ∫2x\sqrt{2x−3}\,dx \quad = \quad \tfrac{2}{5}(1+x)(−3+2x)^{3/2}+C$$
b. $$\displaystyle ∫2x\sqrt{2x−3}\,dx \quad = \quad \tfrac{2}{5}(1+x)(−3+2x)^{3/2}+C$$

In exercises 52 - 57, state whether you would use integration by parts to evaluate the integral. If so, identify $$u$$ and $$dv$$. If not, describe the technique used to perform the integration without actually doing the problem.

### Exercise $$\PageIndex{52}$$

$$\displaystyle ∫x\ln x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{53}$$

$$\displaystyle ∫\frac{\ln^2x}{x}\,dx$$

Do not use integration by parts. Choose $$u$$ to be $$\ln x$$, and the integral is of the form $$\displaystyle ∫u^2\,du.$$

### Exercise $$\PageIndex{54}$$

$$\displaystyle ∫xe^x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{55}$$

$$\displaystyle ∫xe^{x^2−3}\,dx$$

Do not use integration by parts. Let $$u=x^2−3$$, and the integral can be put into the form $$∫e^u\,du$$.

### Exercise $$\PageIndex{56}$$

$$\displaystyle ∫x^2\sin x\,dx$$

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{57}$$

$$\displaystyle ∫x^2\sin(3x^3+2)\,dx$$

Do not use integration by parts. Choose $$u$$ to be $$u=3x^3+2$$ and the integral can be put into the form $$\displaystyle ∫\sin(u)\,du.$$

In exercises 58-59, sketch the region bounded above by the curve, the $$x$$-axis, and $$x=1$$, and find the area of the region. Provide the exact form or round answers to the number of places indicated.

### Exercise $$\PageIndex{58}$$

$$y=2xe^{−x}$$ (Approximate answer to four decimal places.)

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{59}$$

$$y=e^{−x}\sin(πx)$$ (Approximate answer to five decimal places.)

The area under graph is $$0.39535 \, \text{units}^2.$$ In exercises 60 - 61, find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.

### Exercise $$\PageIndex{60}$$

$$y=\sin x,\,y=0,\,x=2π,\,x=3π;$$ about the $$y$$-axis (Express the answer in exact form.)

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{61}$$

$$y=e^{−x}, \,y=0,\,x=−1, \, x=0;$$ about $$x=1$$ (Express the answer in exact form.)

$$V = 2πe \, \text{units}^3$$

### Exercise $$\PageIndex{62}$$

A particle moving along a straight line has a velocity of $$v(t)=t^2e^{−t}$$ after $$t$$ sec. How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.)

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{63}$$

Find the area under the graph of $$y=\sec^3x$$ from $$x=0$$ to $$x=1$$. (Round the answer to two significant digits.)

$$A= 2.05 \, \text{units}^2$$

### Exercise $$\PageIndex{64}$$

Find the area between $$y=(x−2)e^x$$ and the $$x$$-axis from $$x=2$$ to $$x=5$$. (Express the answer in exact form.)

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{65}$$

Find the area of the region enclosed by the curve $$y=x\cos x$$ and the $$x$$-axis for $$\frac{11π}{2}≤x≤\frac{13π}{2}.$$ (Express the answer in exact form.)

$$A = 12π \, \text{units}^2$$

### Exercise $$\PageIndex{66}$$

Find the volume of the solid generated by revolving the region bounded by the curve $$y=\ln x$$, the $$x$$-axis, and the vertical line $$x=e^2$$ about the $$x$$-axis. (Express the answer in exact form.)

Add texts here. Do not delete this text first.

### Exercise $$\PageIndex{67}$$

Find the volume of the solid generated by revolving the region bounded by the curve $$y=4\cos x$$ and the $$x$$-axis, $$\frac{π}{2}≤x≤\frac{3π}{2},$$ about the $$x$$-axis. (Express the answer in exact form.)

$$V = 8π^2 \, \text{units}^3$$
### Exercise $$\PageIndex{68}$$
Find the volume of the solid generated by revolving the region in the first quadrant bounded by $$y=e^x$$ and the $$x$$-axis, from $$x=0$$ to $$x=\ln(7)$$, about the $$y$$-axis. (Express the answer in exact form.)
{"ft_lang_label":"__label__en","ft_lang_prob":0.6821567,"math_prob":0.9999912,"size":12213,"snap":"2019-43-2019-47","text_gpt3_token_len":4479,"char_repetition_ratio":0.319846,"word_repetition_ratio":0.29082915,"special_character_ratio":0.3735364,"punctuation_ratio":0.11372549,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000055,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-17T02:39:04Z\",\"WARC-Record-ID\":\"<urn:uuid:ddc060a0-0749-499c-ad6c-1bbe798797bd>\",\"Content-Length\":\"142169\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2578d648-96d5-4af9-8abc-89ca83755f07>\",\"WARC-Concurrent-To\":\"<urn:uuid:2313df9a-cf79-4c34-aaf8-4a3009541458>\",\"WARC-IP-Address\":\"34.232.212.106\",\"WARC-Target-URI\":\"https://math.libretexts.org/Courses/Mount_Royal_University/MATH_2200%3A_Calculus_for_Scientists_II/2%3A_Techniques_of_Integration/2.1%3A_Integration_by_parts/2.1E%3A_Exercises\",\"WARC-Payload-Digest\":\"sha1:SDOHLJZC5FUJGLHLTJGGUWGP4A5NFLQK\",\"WARC-Block-Digest\":\"sha1:V5TZYL3AHUML72VSQRKXQ6T2O3LCIY5A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496668782.15_warc_CC-MAIN-20191117014405-20191117042405-00025.warc.gz\"}"}