# 6.7E: Exercises for Section 6.7

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

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

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

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

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

In exercises 1 - 3, find the derivative $$\dfrac{dy}{dx}$$.

1) $$y=\ln(2x)$$

$$\dfrac{dy}{dx} = \dfrac{1}{x}$$

2) $$y=\ln(2x+1)$$

3) $$y=\dfrac{1}{\ln x}$$

$$\dfrac{dy}{dx} = −\dfrac{1}{x(\ln x)^2}$$

In exercises 4 - 5, find the indefinite integral.

4) $$\displaystyle ∫\frac{dt}{3t}$$

5) $$\displaystyle ∫\frac{dx}{1+x}$$

$$\displaystyle ∫\frac{dx}{1+x} = \ln|x+1|+C$$

In exercises 6 - 15, find the derivative $$\dfrac{dy}{dx}$$. (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

6) [T] $$y=\dfrac{\ln x}{x}$$

7) [T] $$y=x\ln x$$

$$\dfrac{dy}{dx} = \ln(x)+1$$

8) [T] $$y=\log_{10}x$$

9) [T] $$y=\ln(\sin x)$$

$$\dfrac{dy}{dx} = \cot x$$

10) [T] $$y=\ln(\ln x)$$

11) [T] $$y=7\ln(4x)$$

$$\dfrac{dy}{dx} = \frac{7}{x}$$

12) [T] $$y=\ln\big((4x)^7\big)$$

13) [T] $$y=\ln(\tan x)$$

$$\dfrac{dy}{dx} = \csc x\sec x$$

14) [T] $$y=\ln(\tan 3x)$$

15) [T] $$y=\ln(\cos^2x)$$

$$\dfrac{dy}{dx} = −2\tan x$$

In exercises 16 - 25, find the definite or indefinite integral.

16) $$\displaystyle ∫^1_0\frac{dx}{3+x}$$

17) $$\displaystyle ∫^1_0\frac{dt}{3+2t}$$

$$\displaystyle ∫^1_0\frac{dt}{3+2t} = \tfrac{1}{2}\ln\left(\tfrac{5}{3}\right)$$

18) $$\displaystyle ∫^2_0\frac{x}{x^2+1}\, dx$$

19) $$\displaystyle ∫^2_0\frac{x^3}{x^2+1}\,dx$$

$$\displaystyle ∫^2_0\frac{x^3}{x^2+1}\,dx = 2−\tfrac{1}{2}\ln(5)$$

20) $$\displaystyle ∫^e_2\frac{dx}{x\ln x}$$

21) $$\displaystyle ∫^e_2\frac{dx}{(x\ln x)^2}$$

$$\displaystyle ∫^e_2\frac{dx}{(x\ln x)^2} = \frac{1}{\ln(2)}−1$$

22) $$\displaystyle ∫\frac{\cos x}{\sin x}\, dx$$

23) $$\displaystyle ∫^{π/4}_0\tan x\,dx$$

$$\displaystyle ∫^{π/4}_0\tan x\,dx = \tfrac{1}{2}\ln(2)$$

24) $$\displaystyle ∫\cot(3x)\,dx$$

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

$$\displaystyle ∫\frac{(\ln x)^2}{x}\, dx = \tfrac{1}{3}(\ln x)^3$$

In exercises 26 - 35, compute $$\dfrac{dy}{dx}$$ by differentiating $$\ln y$$.

26) $$y=\sqrt{x^2+1}$$

27) $$y=\sqrt{x^2+1}\sqrt{x^2−1}$$

$$\dfrac{dy}{dx} = \dfrac{2x^3}{\sqrt{x^2+1}\sqrt{x^2−1}}$$

28) $$y=e^{\sin x}$$

29) $$y=x^{−1/x}$$

$$\dfrac{dy}{dx} = x^{−2−(1/x)}(\ln x−1)$$

30) $$y=e^{ex}$$

31) $$y=x^e$$

$$\dfrac{dy}{dx} = ex^{e−1}$$

32) $$y=x^{(ex)}$$

33) $$y=\sqrt{x}\sqrt[3]{x}\sqrt[6]{x}$$

$$\dfrac{dy}{dx} = 1$$

34) $$y=x^{−1/\ln x}$$

35) $$y=e^{−\ln x}$$

$$\dfrac{dy}{dx} = −\dfrac{1}{x^2}$$

In exercises 36 - 40, evaluate by any method.

36) $$\displaystyle ∫^{10}_5\dfrac{dt}{t}−∫^{10x}_{5x}\dfrac{dt}{t}$$

37) $$\displaystyle ∫^{e^π}_1\dfrac{dx}{x}+∫^{−1}_{−2}\dfrac{dx}{x}$$

$$π−\ln(2)$$

38) $$\dfrac{d}{dx}\left[\displaystyle ∫^1_x\dfrac{dt}{t}\right]$$

39) $$\dfrac{d}{dx}\left[\displaystyle ∫^{x^2}_x\dfrac{dt}{t}\right]$$

$$\dfrac{1}{x}$$

40) $$\dfrac{d}{dx}\Big[\ln(\sec x+\tan x)\Big]$$

In exercises 41 - 44, use the function $$\ln x$$. If you are unable to find intersection points analytically, use a calculator.

41) Find the area of the region enclosed by $$x=1$$ and $$y=5$$ above $$y=\ln x$$.

$$(e^5−6)\text{ units}^2$$

42) [T] Find the arc length of $$\ln x$$ from $$x=1$$ to $$x=2$$.

43) Find the area between $$\ln x$$ and the $$x$$-axis from $$x=1$$ to $$x=2$$.

$$\ln(4)−1) \text{ units}^2$$

44) Find the volume of the shape created when rotating this curve from $$x=1$$ to $$x=2$$ around the $$x$$-axis, as pictured here.

45) [T] Find the surface area of the shape created when rotating the curve in the previous exercise from $$x=1$$ to $$x=2$$ around the $$x$$-axis.

$$2.8656 \text{ units}^2$$

If you are unable to find intersection points analytically in the following exercises, use a calculator.

46) Find the area of the hyperbolic quarter-circle enclosed by $$x=2$$ and $$y=2$$ above $$y=1/x.$$

47) [T] Find the arc length of $$y=1/x$$ from $$x=1$$ to $$x=4$$.

$$s = 3.1502$$ units

48) Find the area under $$y=1/x$$ and above the $$x$$-axis from $$x=1$$ to $$x=4$$.

In exercises 49 - 53, verify the derivatives and antiderivatives.

49) $$\dfrac{d}{dx}\Big[\ln(x+\sqrt{x^2+1})\Big]=\dfrac{1}{\sqrt{1+x^2}}$$

50) $$\dfrac{d}{dx}\Big[\ln\left(\frac{x−a}{x+a}\right)\Big]=\dfrac{2a}{(x^2−a^2)}$$

51) $$\dfrac{d}{dx}\Big[\ln\left(\frac{1+\sqrt{1−x^2}}{x}\right)\Big]=−\dfrac{1}{x\sqrt{1−x^2}}$$

52) $$\dfrac{d}{dx}\Big[\ln(x+\sqrt{x^2−a^2})\Big]=\dfrac{1}{\sqrt{x^2−a^2}}$$

53) $$\displaystyle ∫\frac{dx}{x\ln(x)\ln(\ln x)}=\ln|\ln(\ln x)|+C$$