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Mathematics LibreTexts

6.7E: Exercises for Section 6.7

  • Page ID
    70166
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    In exercises 1 - 3, find the derivative \(\dfrac{dy}{dx}\).

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

    Answer:
    \(\dfrac{dy}{dx} = \dfrac{1}{x}\)

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

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

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

     

    In exercises 4 - 5, find the indefinite integral.

    4) \(\displaystyle ∫\frac{dt}{3t}\)

    5) \(\displaystyle ∫\frac{dx}{1+x}\)

    Answer:
    \(\displaystyle ∫\frac{dx}{1+x} = \ln|x+1|+C\)

     

    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] \(y=x\ln x\)

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

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

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

    Answer:
    \(\dfrac{dy}{dx} = \cot x\)

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

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

    Answer:
    \(\dfrac{dy}{dx} = \frac{7}{x}\)

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

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

    Answer:
    \(\dfrac{dy}{dx} = \csc x\sec x\)

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

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

    Answer:
    \(\dfrac{dy}{dx} = −2\tan x\)

     

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

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

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

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

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

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

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

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

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

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

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

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

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

    24) \(\displaystyle ∫\cot(3x)\,dx\)

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

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

     

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

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

    27) \(y=\sqrt{x^2+1}\sqrt{x^2−1}\)

    Answer:
    \(\dfrac{dy}{dx} = \dfrac{2x^3}{\sqrt{x^2+1}\sqrt{x^2−1}}\)

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

    29) \(y=x^{−1/x}\)

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

    30) \(y=e^{ex}\)

    31) \(y=x^e\)

    Answer:
    \(\dfrac{dy}{dx} = ex^{e−1}\)

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

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

    Answer:
    \(\dfrac{dy}{dx} = 1\)

    34) \(y=x^{−1/\ln x}\)

    35) \(y=e^{−\ln x}\)

    Answer:
    \(\dfrac{dy}{dx} = −\dfrac{1}{x^2}\)

     

    In exercises 36 - 40, evaluate by any method.

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

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

    Answer:
    \(π−\ln(2)\)

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

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

    Answer:
    \(\dfrac{1}{x}\)

    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 \(y=\ln x\).

    Answer:
    \((e^5−6)\text{ units}^2\)

    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 \(x=2\).

    Answer:
    \(\ln(4)−1) \text{ units}^2\)

    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.

    This figure is a surface. It has been generated by revolving the curve ln x about the x-axis. The surface is inside of a cube showing it is 3-dimensinal.

    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.

    Answer:
    \(2.8656 \text{ units}^2\)

     

    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 \(x=4\).

    Answer:
    \(s = 3.1502\) 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) \(\displaystyle ∫\frac{dx}{x\ln(x)\ln(\ln x)}=\ln|\ln(\ln x)|+C\)

     

    Contributors

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

     


    6.7E: Exercises for Section 6.7 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.