I use complex analysis to compute the integrals in question.
First, notice that the first integrand is even:
[tex]\dfrac{\cos(-x)}{(-x)^2+1}=\dfrac{\cos(x)}{x^2+1}[/tex]
[tex]\implies\displaystyle\int_0^\infty\frac{\cos(x)}{x^2+1}\,dx=\frac12\int_{-\infty}^\infty\frac{\cos(x)}{x^2+1}\,dx[/tex]
Consider a contour C that's the union of
• Γ, a semicircle of radius R in the upper half-plane, and
• the line segment connecting the points (-R, 0) and (R, 0)
On Γ, we have [tex]z=Re^{it}[/tex] with 0 ≤ t ≤ π.
Consider the complex function
[tex]f(z)=\dfrac{e^{iz}}{z^2+1}[/tex]
and notice that our original integrand is the real part of f(z). Then the integral of f(z) over C is
[tex]\displaystyle\int_Cf(z)\,dz=\lim_{R\to\infty}\left(\int_{-R}^Rf(z)\,dz+\int_\Gamma f(z)\,dz\right)[/tex]
As R → ∞, the first integral on the right is exactly twice the one we want. Estimate the second one to be bounded by
[tex]\displaystyle\left|\int_\Gamma f(z)\,dz\right|\le\pi R|f(z)|\le\frac{\pi R}{R^2-1}[/tex]
since
[tex]|z^2+1|\ge\bigg||z^2|-|-1|\bigg|=|R^2-1|[/tex]
and so the integral along Γ vanishes.
f(z) has only one pole in the interior of C at z = i. By the residue theorem,
[tex]\displaystyle\int_Cf(z)\,dz=2\pi i\,\mathrm{Res}\left(f(z),z=i\right)=2\pi i\lim_{z\to i}(z-i)f(z)=\frac\pi e[/tex]
[tex]\implies\displaystyle\int_0^\infty\frac{\cos(x)}{x^2+1}\,dx=\frac12\mathrm{Re}\left(\int_Cf(z)\,dz\right)=\frac\pi{2e}[/tex]
For the second integral, we recall that for complex z,
[tex]\sqrt z=\exp\left(\dfrac12\left(\ln|z|+i\arg(z)\right)\right)[/tex]
Consider a keyhole contour C, the union of
• [tex]\Gamma_R[/tex], the larger circle with radius R and [tex]z=Re^{it}[/tex], with 0 < t < 2π ;
• [tex]\Gamma_\varepsilon[/tex], the smaller circle with radius ε and [tex]z=\varepsilon e^{-it}[/tex], with 0 < t < 2π ;
• [tex]\ell_1[/tex], the line segment above the positive real axis joining [tex]\Gamma_\varepsilon[/tex] to [tex]\Gamma_R[/tex] ; and
• [tex]\ell_2[/tex], the other line segment below the positive real axis joining [tex]\Gamma_R[/tex] to [tex]\Gamma_\varepsilon[/tex]
Then
[tex]\displaystyle\int_Cf(z)\,dz=\int_{\Gamma_R}f(z)\,dz+\int_{\ell_1}f(z)\,dz+\int_{\Gamma_\varepsilon}f(z)\,dz+\int_{\ell_2}f(z)\,dz[/tex]
and in the limit, the integral over [tex]\ell_1[/tex] converges to the one we want.
Estimate the integrals over the circular arcs:
• [tex]\Gamma_R[/tex] :
[tex]\displaystyle\left|\int_{\Gamma_R}f(z)\,dz\right|\le2\pi R|f(Re^{it})|\le\dfrac{2\pi R^{3/2}}{|R-\sqrt5|^2}\to0[/tex]
as R → ∞.
• [tex]\Gamma_\varepsilon[/tex] :
[tex]\displaystyle\left|\int_{\Gamma_\varepsilon}f(z)\,dz\right|\le2\pi \varepsilon|f(\varepsilon e^{-it})|\le\dfrac{2\pi\varepsilon^{3/2}}{|\varepsilon-\sqrt5|^2}\to0[/tex]
as ε → 0.
Consider the integral over [tex]\ell_2[/tex] :
[tex]\displaystyle\int_{\ell_2}f(z)\,dz=\int_R^\varepsilon\frac{\sqrt z}{z^2+2z+5}\,dz\\\\=\int_R^\varepsilon\frac{\exp\left(\dfrac12\left(\ln|z|+2\pi i\right)\right)}{z^2+2z+5}\,dz\\\\=-\int_R^\varepsilon\frac{\exp\left(\dfrac12\ln|z|\right)}{z^2+2z+5}\,dz\\\\=\int_\varepsilon^R\frac{\sqrt z}{z^2+2z+5}\,dz\\\\=\int_{\ell_1}f(z)\,dz[/tex]
so in fact,
[tex]\displaystyle\int_Cf(z)\,dz=2\int_0^\infty\frac{\sqrt x}{x^2+2x+5}\,dx[/tex]
By the residue theorem,
[tex]\displaystyle\int_Cf(z)\,dz=2\pi i\sum_{\rm poles}\mathrm{Res}\,f(z)[/tex]
We have poles at z = -1 + 2i and z = -1 - 2i. On our chosen branch,
[tex]\sqrt{-1+2i}=i\sqrt[4]{5}\exp\left(-\dfrac i2\tan^{-1}(2)\right)[/tex]
[tex]\sqrt{-1-2i}=i\sqrt[4]{5}\exp\left(\dfrac i2\tan^{-1}(2)\right)[/tex]
The residues are
[tex]\mathrm{Res}(f(z),z=-1-2i)=\dfrac{i\sqrt[4]{5}\exp\left(\frac i2\tan^{-1}(2)\right)}{-4i}[/tex]
[tex]\mathrm{Res}(f(z),z=-1+2i)=\dfrac{i\sqrt[4]{5}\exp\left(-\frac i2\tan^{-1}(2)\right)}{4i}[/tex]
Their sum is
[tex]\displaystyle\sum_{\rm poles}\mathrm{Res}\,f(z)=-\frac{\sqrt[4]{5}}2\sin\left(\dfrac12\tan^{-1}(2)\right)=-\frac{\sqrt[4]{5}}2\sqrt{\frac{5-\sqrt5}{10}}=-\frac i2\sqrt{\frac1\phi}[/tex]
where ɸ = (√5 + 1)/2 is the golden ratio, and so the overall integral is
[tex]\displaystyle\int_0^\infty\frac{\sqrt x}{x^2+2x+5}\,dx=\frac\pi2\sqrt{\frac1\phi}[/tex]
Lastly, recall
[tex]\displaystyle\sum_{k=0}^\infty\frac1{k!}=e[/tex]
Then our expression reduces to
[tex]\left(\dfrac{e\times\frac\pi{2e}}{\frac\pi2\sqrt{\frac1\phi}}\right)^2=\boxed{\phi}[/tex]
someone please help
If x = 3i, y = 2i, and z = m + i, the expression xyz equals
1.) -12-12mi
2.)-6-6mi
3.)12-12mi
4.)6-6mi
Answer:
Step-by-step explanation:
I am assuming that i = √(-1)
3i*2i*(m + i)
= 6i^2(m + i)
= -6(m+ i)
= -6m - 6i.
Line L in the figure below is parallel to the line y=5x+3. Find the coordinates of the point P.
keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above
[tex]y=\stackrel{\stackrel{m}{\downarrow }}{5}x+3\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]
that means that the slope of line L is also 5, or 5/1 or [tex]slope=\cfrac{\stackrel{rise}{5}}{\underset{run}{1}}[/tex]
Check the picture below.
For every 6kg of a child's weight, pediatricians prescribe 17mg of acetaminophen. If Lexie weighs 48kg, how many milligrams of the acetaminophen will the pediatrician prescribe?
HELP ILL MARK BRAINLIEST IF CORRECT!!!
Answer:
Does y vary directly with x?
•YES
What is the---------------?
•A. k=y
What is the function----------?
•A. y=x
Help help math please please
[tex]\underline{\underline{\large\bf{Solution:-}}}\\[/tex]
The standard form of a line is given by -
[tex]\green{ \underline { \boxed{ \sf{Ax+By=C}}}}[/tex]
where
A= integerx = x-interceptB= integery = y-interceptC = constantGiven Equation: -[tex]\begin{gathered}\\\implies\quad \sf y = -\frac{5x}{4} -1 \\\end{gathered} [/tex]
Arranging equation in standard form -
[tex]\begin{gathered}\\\implies\quad \sf y = -\frac{5x}{4} - \frac{1}{1} \quad (1 = \frac{1}{1})\\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf y = -\frac{5x}{4} - \frac{1 \times 4}{1 \times 4} \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf y = -\frac{5x}{4} - \frac{4}{4} \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf y = \frac{-5x-4}{4} \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf 4y = -5x-4 \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf 4y +5x =-4 \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf 5x +4y=-4 \\\end{gathered} [/tex]
Comparing with standard form of line -
A = 5B = 4Thus , 5 is our required answer.
can someone help me with this I already answered half of all of them but I need to make sure It's right?
Answer:
yes 0.265 is correct it is your answer
In the diagram below, xy and yz are tangent to 0. What is the measure of
wz?
W!
0 120°
60°y
Z
A. 240°
B. 180°
C. 210°
D. 150
Answer:
240°
Step-by-step explanation:
measure of XWZ = 360 - 120 = 240°
The measure of arc wz is 240 degrees.
here, we have,
we know that,
A small arc is one with a measure of fewer than 180 degrees. A major arc is one with a measure greater than 180 degrees. A semicircle is an arc that divides a circle in half and has a measure of 180 degrees.
An arc is a continuous segment of a circle. There are two sections, one of which is larger than the other. The major arc is the larger one, and the minor arc is the smaller one.
Given:
From diagram, m ∠xz = 120 degrees.
We know that whole circle = 360 degrees
So, measure of arc wz = 360 - 120
= 240
Hence, the measure of arc wz is 240 degrees.
Learn more about Major and Minor Arc here:
brainly.com/question/9288523
#SPJ7
A market researcher for a consumer electronics company wants to determine if the residents of a particular city are spending more time watching TV than the average for this geographic area. The average for this geographic area is 13 hours per week. A random sample of 16 respondents of the city is selected, and each respondent is instructed to keep a detailed record of all television viewing in a particular week. For this sample the viewing time per week has a mean of 15.3 hours and a sample standard deviation sn = 3.8 hours. Assume that the amount of time of television viewing per week is normally distributed. Can the researcher claim that the residents of this particular city are spending significantly (at 5% level) more time watching TV than the average for this geographic area? Explain your conclusion.
Answer:
The conclusion
There is sufficient evidence to conclude that the the residents of this particular city are spending significantly (at 5% level) more time watching TV than the average for this geographic area.
Step-by-step explanation:
From the question we are told that
The population mean is [tex]\mu = 13 \ hours\ per\ week[/tex]
The sample size is n= 16
The sample mean is [tex]\= x = 15.3[/tex]
The standard deviation is [tex]s = 3.8\ hours.[/tex]
The null hypothesis is [tex]H_o : \mu = 13[/tex]
The alternative hypothesis is [tex]H_a : \mu > 13[/tex]
The level of significance is [tex]\alpha = 0.05[/tex]
Generally the test statistics is mathematically represented as
[tex]t = \frac{\= x - \mu }{ \frac{\sigma }{\sqrt{n} } }[/tex]
=> [tex]t = \frac{15.3 - 13 }{ \frac{ 3.8 }{\sqrt{16} } }[/tex]
=> [tex]t = 2.42[/tex]
Generally the p-value is mathematically represented as
[tex]p-value = P(z > 2.42)[/tex]
From the z -table
[tex]P(z > 2.42) = 0.0077603[/tex]
=> [tex]p-value = 0.0077603[/tex]
From the values obtained we see that [tex]p-value < \alpha[/tex]
Hence we reject the null hypothesis
Therefore there is sufficient evidence to conclude that the the residents of this particular city are spending significantly (at 5% level) more time watching TV than the average for this geographic area.
¿Cuál es la fuerza por unidad de longitud que actúa
sobre un alambre recto que conduce una corriente de
9.40 A de manera perpendicular a un campo
magnético uniforme de 0.80 T?
La fuerza por unidad de longitud del cable recto que lleva es de 7,52 N/m.
¿Qué es la fuerza magnética?La fuerza magnética se refiere a la fuerza que experimenta un cuerpo en el campo magnético.
Tenemos la siguiente información;
Corriente (I) = 9,40 A
Campo magnético = 0,80 T
Dado que;
F = BIL senθ
Tenemos que encontrar la fuerza por unidad de longitud así;
F/L = BIsenθ
F/L = 0,80 T * 9,40 A Sen 90 = 7,52 N/m
Obtenga más información sobre el campo magnético: https://brainly.com/question/19542022
1
Calculate the pressure exerted by a girl of 30kg if she is standing on only one
leg. (Area of her sole is 100 cm).
Soln
here
weight = 30 kg
Force = weight × 10
= 30×10
= 300
Area = cm
= 100/100
= 1
now,
Pressure = Force/Area
= 300/1
= 300 pa.
a car travels at a speed of 85 mph how far will it travel in 15 minutes
Answer:
106.25 mph
Step-by-step explanation:
since the car travels at 85 miles per hour we divide it by 60 seconds in order to find how many miles is it to get in a minute
=> 85mph ÷ 60
divide 85 by 60
=> 1.416
multiply 1.416 by 15 to get how many miles it to go in 15 minutes
=> 21.25
then add it to 85
=> 106.25
The value of square root of 20 is between 4.4 and 4.5. Which of the following is a more precise approximation of square root of 20?
Answer:
4.5
Step-by-step explanation:
the square root of 20 is closer to 4.5 than it is to 4.4
Looking for answer for this answer
Answer:
The last option
Step-by-step explanation:
[tex]a^{-4} =\frac{1}{a^{4} }[/tex]
then
[tex]2a^{-4} m^{5}=\frac{2m^{5} }{a^{4} }[/tex]
Hope this helps
Are the lines x = 2 and y = 5 parallel? Are the perpendicular? Explain.
=
Answer: Yes
Step-by-step explanation:
x=2 is a vertical line with an undefined slope, and y=5 is a horizontal line with a slope of 0. Since these lines have slopes that are negative reciprocals, they are perpendicular.
Which one is greater 0.5 or 1/2
Answer:
Both are equal.
Step-by-step explanation:
.5 is a decimal and you get it by dividing 1 by 2, or 1/2.
So 1/2 ultimately means .5.
please answer these 3 equations fast pls
Answer:
15: 10.99
2(3.14)r
3.5/2 = 1.75
2(3.14)1.75
10.99
16: 50.24
2(3.14)8
50.24
17: 260%
2 = 200%
3/5 = 60%
200 + 60 = 260%
Have an amazing day!
Please rate and mark brainliest!!
2. A car costs $10,350 and decreases in value by 1% per month. How much will the car be worth
after 3 years?
Answer:
The car will be worth 6,624$ in 3-years
Step-by-step explanation:
Firstly we need to move the decimal two to the left, so the decimal form of 1% is 0.01.
Secondly, we make the equation by plugging in a multiplication sign and the cost of the car. Example - (0.01 x 10,350).
Now, that we have everything in an equation we can multiply 0.01 by 10,350, which will give us what 0.01 percent of 10,350 is, (103.5).
Lastly, we will subtract 103.5 from the cost of the car (10,350) to get the value of the car after the first month. The cost of the car will be $10,246.5. To find the value after the next two years we will repeat these steps...
Find the slope of the line that goes through the given points
(4,5) (and 3,3)
Answer:
slope=2
Step-by-step explanation:
y-y=m(4-3)
5-3=m(4-3)
2=m(1)
2/1=m
m=2
if a printer prints 36 pages in 12 minutes, how many pages per minute?
Step-by-step explanation:
The key word "per" hints at a ratio
A ratio is represented by ?:?, ?/?, etc
So, we can divide the two
pages. 36 pages
----------- = ------------- = 3 pages per min minute 12 minutes
Ok, so Jay has a goal of saving money to buy a handheld game console. Jay has reached 10% of his goal and saved $12. Please explain
Answer:
$120
Step-by-step explanation:
10% = 1/10
If Jay has only saved 1/10 of his goal, then to find how much the goal is we multiply how much he has by 10. This will convert 1/10 to 1.
12 x 10 = 120
Jay's goal is $120
Answer:
$120
i got on edg 2020 hope this helped you
NO LINKS!!!
Please help with these parts!! parts b, c, and d
b. There are infinitely many solutions there are infinitely many numbers that are greater than or equal to 4
c. The boundary point for the inequality is at x=4 because any value less would make the inequality not true. The ralationship with the inequality is that it is the only solution that makes the inequality exactly equal to 13
d. (see image) just account for the x direction and ignore the y
A is solved according to you
#b
We have to create a interval notation to find
Let's check
[tex]\\ \rm\rightarrowtail 4x-3\geqslant 13[/tex]
[tex]\\ \rm\rightarrowtail 4x\geqslant 16[/tex]
[tex]\\ \rm\rightarrowtail x\geqslant 4[/tex]
So
[tex]\\ \rm\rightarrowtail x\in [4,\infty)[/tex]
Infinite solutions#c
Boundary is when the inequality becomes false
Here its
x=4#d
Attached
Jessica is on her way home in her car. She has driven 30 miles so far, which is two-thirds of the way home. What is the total length of her drive?
Calculus 3.10 Linearization (a) y=x^2sin2x (b) y=ln sqrt(1+t^2) find the differential of each.
Answer:
Below in bold.
Step-by-step explanation:
(a) y=x^2sin2x
Using the Product Rule:
dy/dx = x^2*2cos2x + 2xsin2x
dy/dx = 2x(xcos2x + sin2x).
(b) y=ln sqrt(1+t^2)
y = ln (1 + t^2)^1/2
Using the chain rule:
dy/dx = 1 / (1 + t^2)^1/2 * 1/2(1 + t^2)^-1/2 * 2t
= 2t/(1 + t^2)^1/2 * 1/ (2(1 + t^2)^1/2
= 2t / 2(1 + t^2)
= t / ( 1 + t^2).
Circumference of the circle
R=9in
Answer:
C = 56.52 Inch.
Step-by-step explanation:
Formula: C = 2πr
C = 2π (9)
C = 18π
C = 18 (3.14)
C = 56.52 Inches
What two factors of 25 also have the sum of -10?
Answer:
-5 and -5.
Step-by-step explanation:
Two negative numbers multiply to make a positive and add to make a negative.
[tex] \frac{x}{4} + 3 = \frac{x}{2} + 3[/tex]
Answer:I think it’s 7...
Step-by-step explanation:
20 m/h
10. An aeroplane travel 2000 miles in 2
hours, what is its speed?
Answer:
1000 mph or 1000 miles per hour
Step-by-step explanation:
Speed = Distance/time, 2000miles/2hours=1000 miles per hour or 1000 mph
A statistician believes that the percent population growth rate of major cities has decreased compared to the last decade. To test his claim, he randomly selects major cities and compares the percent population growth between the years 2007 and 2017 to the percent population growth between the years 1997 and 2007. Suppose that data were collected for a random sample of 17 cities, where each difference is calculated by subtracting the percent growth rate from 1997 to 2007 from the percent growth rate from 2007 to 2017. Assume that the percentages are normally distributed. The statistician uses the alternative hypothesis Ha:μd<0. Using a test statistic of t≈−5.053, which has 16 degrees of freedom, determine the range that contains the p-value.
Answer:
The range consisting of the p-value is, 0.000001 < p-value < 0.00001.
Step-by-step explanation:
The dependent t-test (also known as the paired t-test or paired samples t-test) compares the two means associated groups to conclude if there is a statistically significant difference amid these two means.
In this case a paired t-test is used to determine whether the percent population growth rate of major cities has decreased compared to the last decade.
The alternative hypothesis is Hₐ: [tex]\mu_{d}<0[/tex].
The test statistic value is, t = -5.053.
The degrees of freedom is, 16.
Compute the p-value as follows:
[tex]p-value=P(t_{16}<-5.053)=0.000059[/tex]
Thus, the range consisting of the p-value is, 0.000001 < p-value < 0.00001.
What's the radius of a circle with 15 mm? please answer this question is easy!
Answer:
7.5
Step-by-step explanation:
If Circumference = 15 pi, then the diameter is 15 mm, so the radius = 7.5 mm.