Deck 7: Techniques of Integration

Question 1

Evaluate the integral. $\int_{2}^{\infty} \frac{d y}{y^{2}+2 y-3}$

A) $\frac{5}{2}$
B) $\frac{5}{4}$
C) $\frac{1}{\ln 4}$
D) $\text { divergent }$
E) $\frac{\ln 5}{4}$

Answer 1

$\frac{\ln 5}{4}$

Question 2

Evaluate the integral if it is convergent.

Answer 2

Question 3

Determine whether the improper integral converges or diverges, and if it converges, find its value.

Answer 3

not answered

Question 4

Find the area bounded by the curves $y=\cos x$ and $y=\cos ^{2} x$ between $x=0$ and $x=\frac{\pi}{2}$ .

A) $\frac{1}{4}$
B) $4$
C) $\frac{\sqrt{2}}{2}$
D) $\frac{1}{2}$
E) $\text { none of these }$

Question 5

Let a and b be real numbers. What integral must appear in place of the question mark "?" to make the following statement true? $\int_{-\infty}^{a} \frac{10}{x^{2}+9} d x+\int_{a}^{\infty} \frac{10}{x^{2}+9} d x=?+\int_{b}^{\infty} \frac{10}{x^{2}+9} d x$

A) $\int_{-\infty}^{a} \frac{9}{x^{2}+10} d x$
B) $\int_{0}^{-\infty} \frac{10}{x^{2}-9} d x$
C) $\int_{-\infty}^{b} \frac{10}{x^{2}+9} d x$
D) $\int_{0}^{-\infty} \frac{10}{x^{2}+9} d x$
E) $\text { none of these }$

Question 6

Evaluate the integral or show that it is divergent. $\int_{-\infty}^{\infty} \frac{5 d x}{4 x^{2}+4 x+5}$

A) $-\frac{\pi}{10}$
B) $\frac{\pi}{5}$
C) $\frac{\pi}{8}$
D) $\frac{5}{4}$ $\pi$
E) $\text { divergent }$

Question 7

For what values of K is the following integral improper?

Question 8

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-27}^{8} \frac{1}{\sqrt[3]{x}} d x$

A) $15$
B) $-\frac{15}{2}$
C) Diverges
D) $\frac{15}{2}$

Question 9

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x^{2} \ln x}$

A) $-\frac{1}{4}$
B) $\frac{1}{4}$
C) $\text { divergent }$
D) 2
E) $\frac{1}{2}$

Question 10

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{3 \pi}^{\infty} \cos x d x$

A) 6
B) 0
C) 3
D) Diverges

Question 11

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-\infty}^{\infty} \frac{3 e^{x}}{3+e^{2 x}} d x$

A) $\frac{\pi \sqrt{3}}{2}$
B) $\sqrt{3}$
C) $\frac{\pi \sqrt{3}}{3}$
D) Diverges

Question 12

Determine whether the improper integral converges or diverges, and if it converges, find its value.

Question 13

Determine whether the integral converges or diverges. If it converges, find its value.

Question 14

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{1} \frac{d x}{2 x+4} ; \quad n=7$

A) 0.2029
B) 0.1088
C) 0.1163
D) 0.2326

Question 15

Evaluate the integral.

Question 16

Evaluate the integral or show that it is divergent.

Question 17

A manufacturer of light bulbs wants to produce bulbs that last about $400$ hours but, of course, some bulbs burn out faster than others. Let $F(t)$ be the fraction of the company's bulbs that burn out before t hours. $F(t)$ lies between 0 and 1. Let $r(t)=F^{\prime}(t)$ . What is the value of $\int_{0}^{\infty} r(t) d t$ ?

A) $\int_{0}^{\infty} r(t) d t=400$
B) $\int_{0}^{\infty} r(t) d t=2$
C) $\int_{0}^{\infty} r(t) d t=0$
D) $\text { divergent }$
E) $\int_{0}^{\infty} r(t) d t=1$

Question 18

The region $\left\{(x+y) \mid x \geq-7,0 \leq y \leq e^{-x / 5}\right\}$ is represented below. Find the area of this region to two decimal places.

A) 20.28
B) $17.89$
C) $16.08$
D) $15.89$
E) $15.87$

Question 19

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{2 \pi}^{\infty} \cos x d x$

A) Diverges
B) 0
C) 2
D) 4

Question 20

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{3}^{\infty} \frac{1}{x^{3}} d x$

A) Diverges
B) $\frac{1}{9}$
C) $\frac{1}{18}$
D) 0

Question 21

A body moves along a coordinate line in such a way that its velocity at any time t, where , is given by .
Find its position function if it is initially located at the origin.

Question 22

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals.

Question 23

Use the Table of Integrals to evaluate the integral. $\int e^{4 x} \sin 2 x d x$

A) $-\frac{1}{5} e^{4 x} \sin 2 x+\frac{1}{10} e^{4 x} \cos 2 x+C$
B) $\frac{1}{5} e^{4 x} \sin 2 x+\frac{1}{10} e^{4 x} \cos 2 x+C$
C) $\frac{1}{5} e^{4 x} \sin 2 x-\frac{3}{10} e^{4 x} \cos 2 x+C$
D) $-\frac{1}{5} e^{4 x} \sin 2 x-\frac{1}{10} e^{4 x} \cos 2 x+C$
E) $\frac{1}{5} e^{4 x} \sin 2 x-\frac{1}{10} e^{4 x} \cos 2 x+C$

Question 24

The region under the curve $y=2 \sin ^{2} x$ , $0 \leq x \leq \pi$ is rotated about the x-axis. Find the volume of the resulting solid.

A) $\frac{3}{4}$ $\pi^{2}$
B) $\frac{2}{\pi}$
C) $\frac{6}{\pi^{2}}$
D) $\frac{\pi}{6}$
E) $\frac{\pi^{2}}{6}$

Question 25

Estimate the area of the shaded region by using the Trapezoidal Rule with . Round the answer to the nearest tenth.

Question 26

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals.

Question 27

Use the Trapezoidal Rule to approximate for . Round the result to four decimal places.

Question 28

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value.

Question 29

Use the Midpoint Rule to approximate the given integral with the specified value of n. Compare your result to the actual value. Find the error in the approximation. $2 \int_{2}^{3} e^{-\sqrt{x}} d x, \quad n=6$

A) $-0.00008$
B) $0.60004$
C) 0.00008
D) 1.00008
E) $-0.00096$

Question 30

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{\pi / 2} \sqrt{4+\sin ^{2} x} d x ; \quad n=6$

A) 4.0689
B) 3.3296
C) 2.9599
D) 3.6993

Question 31

Use a table of integrals to evaluate the integral. $\int x^{3} \sin \left(x^{2}+3\right) d x$

A) $\frac{1}{2} \sin \left(x^{2}+\sqrt{3}\right)-\frac{1}{2} x^{2} \cos \left(x^{2}+\sqrt{3}\right)+C$
B) $\frac{1}{2} \sin \left(x^{2}+3\right)-\frac{1}{2} x^{2} \cos \left(x^{2}+3\right)+C$
C) $-\frac{1}{2} \sin \left(x^{2}+\sqrt{3}\right)-\frac{1}{2} x^{2} \cos \left(x^{2}+\sqrt{3}\right)+C$
D) $-\frac{1}{2} \sin \left(x^{2}+3\right)-\frac{1}{2} x^{2} \cos \left(x^{2}+3\right)+C$

Question 32

Use a table of integrals to evaluate the integral. $\int e^{-7 x} \sin 3 x d x$

A) $-\frac{1}{10} e^{-7 x}(7 \sin 3 x+7 \cos 3 x)+C$
B) $-\frac{1}{10} e^{-7 x}(7 \sin 3 x+3 \cos 3 x)+C$
C) $-\frac{1}{58} e^{-7 x}(7 \sin 3 x+7 \cos 3 x)+C$
D) $-\frac{1}{58} e^{-7 x}(7 \sin 3 x+3 \cos 3 x)+C$

Question 33

Use a table of integrals to evaluate the integral. $\int x \sqrt{2+2 x} d x$

A) $\frac{2 \sqrt{2}}{15}(2 x-1)(x+1)^{3 / 2}+C$
B) $\frac{2 \sqrt{2}}{15}(3 x-2)(x+1)^{3 / 2}+C$
C) $\frac{2}{15}(3 x-2) \sqrt{x+1}+C$
D) $\frac{2 \sqrt{2}}{15}(3 x-2) \sqrt{x+1}+C$

Question 34

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals.

Question 35

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with and the formula to estimate the person's cardiac output, where D is the quantity of dye injected in milligrams, is the concentration of the dye in the aorta, and R is measured in liters per minute. Round to one decimal place.
t
0
2
4
6
8
10
12
14
16
18
20
22
24
C(t)
0
0
2.6
6.3
9.7
7.5
4.5
3.5
2.2
0.6
0.3
0.1
0

Question 36

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{2} \frac{d x}{\sqrt{x^{3}+4}} ; \quad n=6$

A) 0.8528
B) 0.9842
C) 0.4921
D) 0.4695

Question 37

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{\pi / 2} \sqrt{2+\sin ^{2} x} d x ; \quad n=6$

A) 2.2028
B) 2.7519
C) 3.0265
D) 2.4774

Question 38

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with subintervals.

Question 39

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{-1}^{1} \sqrt{x^{2}+1} d x ; \quad n=6$

A) 2.0076
B) 2.9504
C) 2.6098
D) 2.2955

Question 40

Use the Table of Integrals to evaluate the integral. $\int \frac{\sqrt{49 x^{2}-1}}{x^{2}} d x$

A) $7 \ln \left|7 x+\sqrt{49 x^{2}-1}\right|+C$
B) $-\frac{\sqrt{49 x^{2}-1}}{x}+7 \ln \left|14 x+2 \sqrt{49 x^{2}-1}\right|+C$
C) $\frac{\sqrt{49 x^{2}-1}}{x}+7 \ln \left|7 x+\sqrt{49 x^{2}-1}\right|$
D) $\frac{\sqrt{7 x^{2}-1}}{x}+\ln \left|x+\sqrt{7 x^{2}-1}\right|+C$
E) $\frac{\sqrt{49 x^{2}-1}}{x}+\ln \left|7 x+\sqrt{49 x^{2}}\right|+C$

Question 41

Use the Table of Integrals to evaluate the integral to three decimal places.

Question 42

Evaluate the integral. $\int_{x / 6}^{\pi / 3} \frac{3 \ln (\tan x)}{7 \sin x \cos x} d x$

A) $-\frac{1}{5}(\ln 2)^{2}$
B) $-\frac{1}{2}(\ln 3)^{2}$
C) 0
D) $\frac{1}{6}(\ln 3)^{2}$
E) $\frac{1}{2}(\ln 3)^{2}$

Question 43

Evaluate the integral.

Question 44

Evaluate the integral. $\int \frac{1}{-e^{-5 x}+e^{5 x}} d x$

A) $-\frac{1}{10} \ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$
B) $\frac{1}{10} \ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$
C) $\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$
D) $-\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}}\right)+C$
E) $-\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$

Question 45

Evaluate the integral.

Question 46

Write the form of the partial fraction decomposition of the rational expression. Do not find the numerical values of the constants. $\frac{x^{2}-x-8}{2 x^{3}-3 x^{2}+4 x-6}$

A) $\frac{A}{2 x-3}+\frac{B x+C}{x^{2}+2}$
B) $\frac{A}{2 x-1}+\frac{B x+C}{x^{2}+2}$
C) $\frac{A}{2 x+1}+\frac{B x+C}{x^{2}-2}$
D) $\frac{A}{2 x+3}+\frac{B x+C}{x^{2}-2}$

Question 47

Evaluate the integral.

Question 48

Find the integral. $\int \frac{x^{3}-3 x^{2}+6 x-2}{x^{3}-2 x^{2}+x} d x$

A) $\ln \left|\frac{x-1}{x^{2}}\right|-\frac{1}{x-1}+x+C$
B) $\ln \left|\frac{x^{2}}{x-1}\right|-\frac{2}{x-1}+x+C$
C) $\ln \left|\frac{x-1}{x^{2}}\right|-\frac{2}{x-1}+x+C$
D) $\ln \left|\frac{x^{2}}{x-1}\right|-\frac{1}{x-1}+x+C$

Question 49

Use long division to evaluate the integral. $\int \frac{x^{2}}{x+3} d x$

A) $\frac{1}{2}(x-9)(x+3)+9 \ln |x+3|+C$
B) $\frac{1}{2}(x+9)(x+3)-9 \ln |x+3|+C$
C) $\frac{x^{2}}{2}-6 x-27+\ln |x+3|+C$
D) $\frac{x^{2}}{2}-6 x+27+9 \ln |x+3|+C$
E) $\frac{x^{2}}{2}+2 x+\ln |x+9|+C$

Question 50

Evaluate the integral.

Question 51

Find the integral. $\int \frac{3 x-5}{x^{2}-2 x-3} d x$

A) $\ln \left|(x-3)(x+1)^{2}\right|+C$
B) $\ln \left|(x+3)(x-1)^{2}\right|+C$
C) $\ln \left|(x+1)(x-3)^{2}\right|+C$
D) $\ln \left|(x-1)(x+3)^{2}\right|+C$

Question 52

Use the Table of Integrals to evaluate the integral.

Question 53

Evaluate the integral.

Question 54

Find the integral. $\int \frac{d x}{x(x-2)}$

A) $2 \ln \left|\frac{x^{2}-2}{x}\right|+C$
B) $2 \ln \left|\frac{x-2}{x}\right|+C$
C) $\frac{1}{2} \ln \left|\frac{x^{2}-2}{x}\right|+C$
D) $\frac{1}{2} \ln \left|\frac{x-2}{x}\right|+C$

Question 55

Evaluate the integral.

Question 56

Find the integral. $\int \frac{5 x^{2}-9 x+6}{x^{3}-2 x^{2}+x} d x$

A) $\ln \left|\frac{x^{6}}{x-1}\right|-\frac{1}{x-1}+C$
B) $\ln \left|\frac{x-1}{x^{6}}\right|-\frac{1}{x-1}+C$
C) $\ln \left|\frac{x-1}{x^{6}}\right|-\frac{2}{x-1}+C$
D) $\ln \left|\frac{x^{6}}{x-1}\right|-\frac{2}{x-1}+C$

Question 57

Use a table of integrals to evaluate the integral. $\int x \sqrt{2+2 x} d x$

A) $\frac{2 \sqrt{2}}{15}(3 x-2) \sqrt{x+1}+C$
B) $\frac{2 \sqrt{2}}{15}(2 x-1)(x+1)^{3 / 2}+C$
C) $\frac{2}{15}(3 x-2) \sqrt{x+1}+C$
D) $\frac{2 \sqrt{2}}{15}(3 x-2)(x+1)^{3 / 2}+C$

Question 58

Evaluate the integral. $\int \frac{7 d x}{\left(x^{2}+2 x+2\right)^{2}}$

A) $\frac{1}{2}\left(\tan ^{-1}(x+2)+\frac{7}{x^{2}+2}\right)+C$
B) $\frac{1}{2}\left(\tan ^{-1}(x+7)+\frac{1}{x^{2}+2 x+2}\right)+C$
C) $\frac{7}{2}\left(\tan ^{-1}(x+1)+\frac{x+1}{x^{2}+2 x+2}\right)+C$
D) $\frac{1}{2}\left(\tan (x+1)+\frac{17}{x^{2}+2 x+2}\right)+C$
E) $\frac{7}{2}\left(\tan (x+2)+\frac{x+1}{x^{2}+2 x+2}\right)+C$

Question 59

Evaluate the integral.

Question 60

Use a table of integrals to evaluate the integral. $\int \frac{\sqrt{4 x+5}}{x^{2}} d x$

A) $-\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$
B) $-\frac{\sqrt{4 x+5}}{x}+\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$
C) $-\frac{\sqrt{4 x+5}}{x}+\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$
D) $-\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$

Question 61

Find the integral. $\int \frac{3 x-3}{x^{2}-x-2} d x$

A) $\ln \left|(x+2)(x-1)^{2}\right|+C$
B) $\ln \left|(x-2)(x+1)^{2}\right|+C$
C) $\ln \left|(x+1)(x-2)^{2}\right|+C$
D) $\ln \left|(x-1)(x+2)^{2}\right|+C$

Question 62

Find the integral using an appropriate trigonometric substitution. $\int \frac{1}{x^{2} \sqrt{x^{2}+25}} d x$

A) $-\frac{\sqrt{x^{2}+25}}{5 x}+C$
B) $\frac{\sqrt{x^{2}+25}}{5 x}+C$
C) $\frac{\sqrt{x^{2}+25}}{25 x}+C$
D) $-\frac{\sqrt{x^{2}+25}}{25 x}+C$

Question 63

Make a substitution to express the integrand as a rational function and then evaluate the integral. Round the answer to four decimal places.

Question 64

Evaluate the integral using the indicated trigonometric substitution.

Question 65

Find the integral using an appropriate trigonometric substitution. $\int \frac{x^{3}}{\sqrt{x^{2}+36}} d x$

A) $\frac{1}{3}\left(x^{2}-36\right)^{3 / 2} \sqrt{x^{2}+36}+C$
B) $\frac{1}{3}\left(x^{2}+72\right) \sqrt{x^{2}+36}+C$
C) $\frac{1}{3}\left(x^{2}-72\right) \sqrt{x^{2}+36}+C$
D) $\frac{1}{3}\left(x^{2}+36\right)^{3 / 2} \sqrt{x^{2}+36}+C$

Question 66

Evaluate the integral. $\int 8\left(\frac{x-1}{x^{2}+2 x}\right) d x$

A) $8\left(\frac{1}{3} x-\frac{1}{3} \ln x\right)+C$
B) $-8(\ln x-x)+C$
C) $8\left(\frac{3}{2} \ln (x+2)-\frac{1}{2} \ln x\right)+C$
D) $-8\left(\frac{\sqrt{2}}{2} \arctan x\right)+C$
E) $\text { None of these }$

Question 67

Find the integral.

Question 68

Find the volume of the resulting solid if the region under the curve from to is rotated about the x-axis. Round your answer to four decimal places.

Question 69

Find the integral using an appropriate trigonometric substitution. $\int \frac{x}{\sqrt{4-x^{2}}} d x$

A) $-\sqrt{4-x^{2}}+C$
B) $\sqrt{2-x}+C$
C) $\sqrt{4-x^{2}}+C$
D) $-\sqrt{2-x}+C$

Question 70

Evaluate the integral using the indicated trigonometric substitution. $\int \frac{x^{3}}{\sqrt{x^{2}+16}} d x ; x=4 \tan \theta$

A) $\frac{3}{2}(x+16)^{3 / 2}-16 \sqrt{x+16}+C$
B) $\left(x^{2}+16\right)^{3 / 2}-\sqrt{x^{2}+16}+C$
C) $\frac{1}{3}\left(x^{2}+16\right)^{3 / 2}-\sqrt{x^{2}+16}+C$
D) $\frac{1}{3}\left(x^{2}+16\right)^{3 / 2}+16 \sqrt{x^{2}+16}+C$
E) $\left(x^{2}+16\right)^{3 / 2}-4 \sqrt{x^{2}+16}+C$

Question 71

Use long division to evaluate the integral. $\int_{0}^{1} \frac{x^{3}+4 x^{2}-12 x+1}{x^{2}+4 x-12} d x$ The choices are rounded to 3 decimal places.

A) $-4.606$
B) $5.394$
C) $0.394$
D) $-9.606$
E) $10.394$

Question 72

Find the integral.

Question 73

The region under the graph of on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

Question 74

A corporation is building a complex of homes, offices, stores, schools, and churches in a rural community. As a result of this development, the planners have estimated that the community's population (in thousands) t years from now will be given by .
What will the average population of the community be over the next 10 years?

Question 75

Evaluate the integral using an appropriate trigonometric substitution. $\int_{0}^{\sqrt{2}} \frac{x^{2}}{\sqrt{4-x^{2}}} d x$

A) $\pi$
B) $\frac{\pi}{2}-1$
C) $\frac{2 \pi}{3}-\frac{\sqrt{3}}{2}$
D) $\pi-1$

Question 76

Evaluate the integral. $\int_{0}^{1} \frac{x}{x+8} d x$

A) $1-9 \ln 8+\sqrt{8} \ln 16$
B) $\frac{1}{8 \ln 9}-\ln \sqrt{8}$
C) $1-8(\ln 9-\ln 8)$
D) $9-8 \ln 9+\ln \sqrt{8}$
E) $\text { None of these }$

Question 77

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt{9-x^{2}} d x$

A) $\frac{1}{3}\left(9-x^{2}\right)^{3 / 2}+C$
B) $-\frac{1}{3} x^{2}\left(9-x^{2}\right)^{3 / 2}+C$
C) $\frac{1}{3} x^{2}\left(9-x^{2}\right)^{3 / 2}+C$
D) $-\frac{1}{3}\left(9-x^{2}\right)^{3 / 2}+C$

Question 78

Evaluate the integral.

Question 79

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with and the formula to estimate the person's cardiac output, where D is the quantity of dye injected in milligrams, is the concentration of the dye in the aorta, and R is measured in liters per minute. Round your answer to one decimal place.
t
0
2
4
6
8
10
12
14
16
18
20
22
24
C(t)
0
0
2.6
5.9
9.7
7.9
4.6
3.5
2.2
0.8
0.2
0.1
0

Question 80

Find the integral.