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Mathematics LibreTexts

Chapter 7 Review Exercises

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    For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

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

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

    Solution: False

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

    4) Integration by parts can always yield the integral.

    Solution: False

    For the following exercises, evaluate the integral using the specified method.

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

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

    Solution: \(\displaystyle −\frac{\sqrt{x^2+16}}{16x}+C\)

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

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

    Solution: \(\displaystyle \frac{1}{10}(4ln(2−x)+5ln(x+1)−9ln(x+3))+C\)

    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

    Soution: \(\displaystyle −\frac{\sqrt{4−sin^2(x)}}{sin(x)}−\frac{x}{2}+C\)

    For the following exercises, integrate using whatever method you choose.

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

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

    Solution: \(\displaystyle \frac{1}{15}(x^2+2)^{3/2}(3x^2−4)+C\)

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

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

    Solution: \(\displaystyle \frac{1}{16}ln(\frac{x^2+2x+2}{x^2−2x+2})−\frac{1}{8}tan^{−1}(1−x)+\frac{1}{8}tan^{−1}(x+1)+C\)

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

    For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

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

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

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

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

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

    For the following exercises, evaluate the integrals, if possible.

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

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

    Solution: approximately 0.2194

    For the following exercises, 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 \(\displaystyle a\) 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 \(\displaystyle v(t)=aexp^{bx}sin(cx)+d.\) (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.

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