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3.4E: Exercises for Section 3.4

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    196032
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax

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    Simplify the expressions in exercises 1 - 5 by writing each one using a single trigonometric function.

    1) \(4−4\sin^2θ\)

    2) \(9\sec^2θ−9\)

    Answer
    \(9\sec^2θ−9 \quad = \quad 9\tan^2θ\)

    3) \(a^2+a^2\tan^2θ\)

    4) \(a^2+a^2\sin^2θ\)

    Answer
    \(a^2+a^2\sin^2θ \quad = \quad a^2\cos^2θ\)

    5) \(16\cos^2θ−16\)

    Use the technique of completing the square to express each trinomial in exercises 6 - 8 as the square of a binomial.

    6) \(4x^2−4x+1\)

    Answer
    \( 4(x−\frac{1}{2})^2\)

    7) \(2x^2−8x+3\)

    8) \(−x^2−2x+4\)

    Answer
    \( −(x+1)^2+5\)

    In exercises 9 - 28, integrate using the method of trigonometric substitution. Express the final answer in terms of the original variable.

    9) \(\displaystyle ∫\frac{dx}{\sqrt{4−x^2}}\)

    10) \(\displaystyle ∫\frac{dx}{\sqrt{x^2−a^2}}\)

    Answer
    \(\displaystyle ∫\frac{dx}{\sqrt{x^2−a^2}} \quad = \quad \ln∣x+\sqrt{−a^2+x^2}∣+C\)

    11) \(\displaystyle ∫\sqrt{4−x^2}\,dx\)

    12) \(\displaystyle ∫\frac{dx}{\sqrt{1+9x^2}}\)

    Answer
    \(\displaystyle ∫\frac{dx}{\sqrt{1+9x^2}} \quad = \quad \tfrac{1}{3}\ln∣\sqrt{9x^2+1}+3x∣+C\)

    13) \(\displaystyle ∫\frac{x^2\,dx}{\sqrt{1−x^2}}\) (Note: You might find the double-angle identity \(\sin(2\theta)=2\sin(\theta)\cos(\theta)\) helpful when converting back to \(x\))

    14) \(\displaystyle ∫\frac{dx}{x^2\sqrt{1−x^2}}\)

    Answer
    \(\displaystyle ∫\frac{dx}{x^2\sqrt{1−x^2}} \quad = \quad −\frac{\sqrt{1−x^2}}{x}+C\)

    15) \(\displaystyle ∫\frac{dx}{(1+x^2)^2}\)

    16) \(\displaystyle ∫\sqrt{x^2+9}\,dx\)

    Answer
    \(\displaystyle ∫\sqrt{x^2+9}\,dx \quad = \quad 9\left[\frac{x\sqrt{x^2+9}}{18}+\tfrac{1}{2}\ln\left|\frac{\sqrt{x^2+9}}{3}+\frac{x}{3}\right|\right]+C\)

    17) \(\displaystyle ∫\frac{\sqrt{x^2−25}}{x}\,dx\)

    18) \(\displaystyle ∫\frac{θ^3}{\sqrt{9−θ^2}}\,dθ\)

    Answer
    \(\displaystyle ∫\frac{θ^3dθ}{\sqrt{9−θ^2}}\,dθ \quad = \quad −\tfrac{1}{3}\sqrt{9−θ^2}(18+θ^2)+C\)

    19) \(\displaystyle ∫\frac{dx}{\sqrt{x^6−x^2}}\)

    20) \(\displaystyle ∫\sqrt{x^6−x^8}\,dx\)

    Answer
    \(\displaystyle ∫\sqrt{x^6−x^8}\,dx \quad = \quad \frac{(−1+x^2)(2+3x^2)\sqrt{x^6−x^8}}{15x^3}+C\)

    21) \(\displaystyle ∫\frac{dx}{(1+x^2)^{3/2}}\)

    22) \(\displaystyle ∫\frac{dx}{(x^2−9)^{3/2}}\)

    Answer
    \(\displaystyle ∫\frac{dx}{(x^2−9)^{3/2}} \quad = \quad −\frac{x}{9\sqrt{x^2-9}}+C\)

    23) \(\displaystyle ∫\frac{\sqrt{1+x^2}}{x}\,dx\)

    24) \(\displaystyle ∫\frac{x^2}{\sqrt{x^2−1}}\,dx\)

    Answer
    \(\displaystyle ∫\frac{x^2}{\sqrt{x^2−1}}\,dx \quad = \quad \tfrac{1}{2}(\ln∣x+\sqrt{x^2−1}∣+x\sqrt{x^2−1})+C\)

    25) \(\displaystyle ∫\frac{x^2}{x^2+4}\,dx\)

    26) \(\displaystyle ∫\frac{dx}{x^2\sqrt{x^2+1}}\)

    Answer
    \(\displaystyle ∫\frac{dx}{x^2\sqrt{x^2+1}} \quad = \quad −\frac{\sqrt{1+x^2}}{x}+C\)

    27) \(\displaystyle ∫\frac{x^2}{\sqrt{1+x^2}}\,dx\)

    28) \(\displaystyle ∫^1_{−1}(1−x^2)^{3/2}\,dx\)

    Answer
    \(\displaystyle ∫^1_{−1}(1−x^2)^{3/2}\,dx \quad = \quad \tfrac{1}{8}\left(x(5−2x^2)\sqrt{1−x^2}+3\arcsin x\right)+C\)

    Use the technique of completing the square to evaluate the integrals in exercises 29 - 33.

    29) \(\displaystyle ∫\frac{1}{x^2−6x}\,dx\)

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

    Answer
    \(\displaystyle ∫\frac{1}{x^2+2x+1}\,dx \quad = \quad −\frac{1}{1+x}+C\)

    31) \(\displaystyle ∫\frac{1}{\sqrt{−x^2+2x+8}}\,dx\)

    32) \(\displaystyle ∫\frac{1}{\sqrt{−x^2+10x}}\,dx\)

    Answer
    \(\displaystyle ∫\frac{1}{\sqrt{−x^2+10x}}\,dx \quad = \quad \arcsin\left( \frac{x-5}{5}\right)+C\)

    33) \(\displaystyle ∫\frac{1}{\sqrt{x^2+4x−12}}\,dx\)

    34) Evaluate the integral without using calculus: \(\displaystyle ∫^3_{−3}\sqrt{9−x^2}\,dx.\)

    Answer
    \(\displaystyle ∫^3_{−3}\sqrt{9−x^2}\,dx \quad = \quad \frac{9π}{2}\); area of a semicircle with radius 3

    35) Find the area enclosed by the ellipse \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1.\)

    36) Evaluate the integral \(\displaystyle ∫\frac{dx}{\sqrt{1−x^2}}\) using two different substitutions. First, let \(x=\cos θ\) and evaluate using trigonometric substitution. Second, let \(x=\sin θ\) and use trigonometric substitution. Are the answers the same?

    Answer
    \(\displaystyle ∫\frac{dx}{\sqrt{1−x^2}} \quad = \quad \arcsin(x)+C\) is the common answer.

    37) Evaluate the integral \(\displaystyle ∫\frac{dx}{x\sqrt{x^2−1}}\) using the substitution \(x=\sec θ\). Next, evaluate the same integral using the substitution \(x=\csc θ.\) Show that the results are equivalent.

    38) Evaluate the integral \(\displaystyle ∫\frac{x}{x^2+1}\,dx\) using the form \(\displaystyle ∫\frac{1}{u}\,du\). Next, evaluate the same integral using \(x=\tan θ.\) Are the results the same?

    Answer
    \(\displaystyle ∫\frac{x}{x^2+1}\,dx \quad = \quad \frac{1}{2}\ln(1+x^2)+C\) is the result using either method.

    39) State the method of integration you would use to evaluate the integral \(\displaystyle ∫x\sqrt{x^2+1}\,dx.\) Why did you choose this method?

    40) State the method of integration you would use to evaluate the integral \(\displaystyle ∫x^2\sqrt{x^2−1}\,dx.\) Why did you choose this method?

    Answer
    Use trigonometric substitution. Let \(x=\sec(θ).\)

    41) Evaluate \(\displaystyle ∫^1_{−1}\frac{x}{x^2+1}\,dx\)

    42) Find the length of the arc of the curve over the specified interval: \(y=\ln x,\quad [1,5].\) Round the answer to three decimal places.

    Answer
    \( s = 4.367\) units

    43) Find the surface area of the solid generated by revolving the region bounded by the graphs of \(y=x^2,\, y=0,\, x=0\), and \(x=\sqrt{2}\) about the \(x\)-axis. (Round the answer to three decimal places).

    44) The region bounded by the graph of \(f(x)=\dfrac{1}{1+x^2}\) and the \(x\)-axis between \(x=0\) and \(x=1\) is revolved about the \(x\)-axis. Find the volume of the solid that is generated.

    Answer
    \( V = \left(\frac{π^2}{8}+\frac{π}{4}\right) \, \text{units}^3\)

    45) Find the area bounded by \(y=\dfrac{2}{\sqrt{64−4x^2}},\, x=0,\, y=0\), and \(x=2\).

    46) An oil storage tank can be described as the volume generated by revolving the area bounded by \(y=\dfrac{16}{\sqrt{64+x^2}},\, x=0,\, y=0,\, x=2\) about the \(x\)-axis. Find the volume of the tank (in cubic meters).

    Answer
    \(V = 24.6\) m3

    47) During each cycle, the velocity \(v\) (in feet per second) of a robotic welding device is given by \(v=2t−\dfrac{14}{4+t^2}\), where \(t\) is time in seconds. Find the expression for the displacement \(s\) (in feet) as a function of \(t\) if \(s=0\) when \(t=0\).

    48) Find the length of the curve \(y=\sqrt{16−x^2}\) between \(x=0\) and \(x=2\).

    Answer
    \( s = \frac{2π}{3}\) units

    49) Use ChatGPT (or another LLM) to analyze when trigonometric substitution works for evaluating integrals, when it does not work, and how to choose the substitution when it does work. Submit your prompt(s), a summary of the output from ChatGPT, and your analysis of how accurate this output is based on the classwork and homework.

    This page titled 3.4E: Exercises for Section 3.4 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.