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Mathematics LibreTexts

Chapter 7 Review Exercises

  • Page ID
    10777
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    In exercises 1 - 4, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

    1) \(\displaystyle ∫e^x\sin(x)\,dx\) cannot be integrated by parts.

    2) \(\displaystyle ∫\frac{1}{x^4+1}\,dx\) cannot be integrated using partial fractions.

    Answer:
    False

    3) In numerical integration, increasing the number of points decreases the error.

    4) Integration by parts can always yield the integral.

    Answer:
    False

     

    In exercises 5 - 10, evaluate the integral using the specified method.

    5) \(\displaystyle ∫x^2\sin(4x)\,dx,\) using integration by parts

    6) \(\displaystyle ∫\frac{1}{x^2\sqrt{x^2+16}}\,dx,\) using trigonometric substitution

    Answer:
    \(\displaystyle ∫\frac{1}{x^2\sqrt{x^2+16}}\,dx = −\frac{\sqrt{x^2+16}}{16x}+C\)

    7) \(\displaystyle ∫\sqrt{x}\ln x\,dx,\) using integration by parts

    8) \(\displaystyle ∫\frac{3x}{x^3+2x^2−5x−6}\,dx,\) using partial fractions

    Answer:
    \(\displaystyle ∫\frac{3x}{x^3+2x^2−5x−6}\,dx = \frac{1}{10}\big(4\ln|2−x|+5\ln|x+1|−9\ln|x+3|\big)+C\)

    9) \(\displaystyle ∫\frac{x^5}{(4x^2+4)^{5/2}}\,dx,\) using trigonometric substitution

    10) \(\displaystyle ∫\frac{\sqrt{4−\sin^2(x)}}{\sin^2(x)}\cos(x)\,dx,\) using a table of integrals or a CAS

    Answer:
    \(\displaystyle ∫\frac{\sqrt{4−\sin^2(x)}}{\sin^2(x)}\cos(x)\,dx = −\frac{\sqrt{4−\sin^2(x)}}{\sin(x)}−\frac{x}{2}+C\)

     

    In exercises 11 - 15, integrate using whatever method you choose.

    11) \(\displaystyle ∫\sin^2 x\cos^2 x\,dx\)

    12) \(\displaystyle ∫x^3\sqrt{x^2+2}\,dx\)

    Answer:
    \(\displaystyle ∫x^3\sqrt{x^2+2}\,dx = \frac{1}{15}(x^2+2)^{3/2}(3x^2−4)+C\)

    13) \(\displaystyle ∫\frac{3x^2+1}{x^4−2x^3−x^2+2x}\,dx\)

    14) \(\displaystyle ∫\frac{1}{x^4+4}\,dx\)

    Answer:
    \(\displaystyle ∫\frac{1}{x^4+4}\,dx = \frac{1}{16}\ln(\frac{x^2+2x+2}{x^2−2x+2})−\frac{1}{8}\tan^{−1}(1−x)+\frac{1}{8}\tan^{−1}(x+1)+C\)

    15) \(\displaystyle ∫\frac{\sqrt{3+16x^4}}{x^4}\,dx\)

     

    In exercises 16 - 18, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

    16) [T] \(\displaystyle ∫^2_1\sqrt{x^5+2}\,dx\)

    Answer:
    \(M_4=3.312,\)
    \(T_4=3.354,\)
    \(S_4=3.326\)

    17) [T] \(\displaystyle ∫^{\sqrt{π}}_0e^{−\sin(x^2)}\,dx\)

    18) [T] \(\displaystyle ∫^4_1\frac{\ln(1/x)}{x}\,dx\)

    Answer:
    \(M_4=−0.982,\)
    \(T_4=−0.917,\)
    \(S_4=−0.952\)

     

    In exercises 19 - 20, evaluate the integrals, if possible.

    19) \(\displaystyle ∫^∞_1\frac{1}{x^n}\,dx,\) for what values of \(n\) does this integral converge or diverge?

    20) \(\displaystyle ∫^∞_1\frac{e^{−x}}{x}\,dx\)

    Answer:
    approximately 0.2194

     

    In exercises 21 - 22, consider the gamma function given by \(\displaystyle Γ(a)=∫^∞_0e^{−y}y^{a−1}\,dy.\)

    21) Show that \(\displaystyle Γ(a)=(a−1)Γ(a−1).\)

    22) Extend to show that \(\displaystyle Γ(a)=(a−1)!,\) assuming \(a\) is a positive integer.

     

    The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity.

    This figure has a graph in the first quadrant. It increases to where x is approximately 03:00 mm:ss and then drops off steep. The maximum height of the graph, here the drop occurs is approximately 420 km/h.

    23) [T] Use the graph to estimate the velocity every 20 sec and fit to a graph of the form \(v(t)=ae^{bx}\sin(cx)+d.\) (Hint: Consider the time units.)

    24) [T] Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.

    Answer:
    Answers may vary. Ex: \(9.405\) km

     

    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.