Let
[tex]I(a) = \displaystyle \int_0^\infty \frac{\sqrt x \arctan(ax)}{1+x^2} \, dx[/tex]
Differentiate with respect to a :
[tex]I'(a) = \displaystyle a \int_0^\infty \frac{x^{\frac32}}{(1+x^2)(1+a^2x^2)} \, dx[/tex]
Substitute y = √x :
[tex]I'(a) = \displaystyle 2a \int_0^\infty \frac{y^4}{(1+y)(1+a^2y)} \, dy[/tex]
Polynomial division yields
[tex]\dfrac{y^4}{(1+y)(1+a^2y)} \\\\ = \dfrac1{a^2}y^2 - \left(\dfrac1{a^2} + \dfrac1{a^4}\right)y + \dfrac1{a^2} + \dfrac1{a^4} + \dfrac1{a^6} - \dfrac{\left(1+\frac1{a^2} + \frac1{a^4} + \frac1{a^6}\right)y + \frac1{a^2} + \frac1{a^4} + \frac1{a^6}}{(1+y)(1+a^2y)}[/tex]
Computing I'(a) isn't so difficult from here. You'd find (assuming a ≥ 0)
[tex]I'(a) = \displaystyle \frac\pi{\sqrt2\left(\sqrt a + a + a^{\frac32} + a^2\right)}[/tex]
Integrate both sides with respect to a. On the right side, substituting b = √a yields
[tex]\displaystyle \int \frac{da}{\sqrt a + a + a^{\frac32} + a^2} = \int \frac{2b}{b + b^2 + b^3 + b^4} \, db \\\\ = 2 \int \frac{db}{1 + b + b^2 + b^3} \\\\ = 2 \int \frac{db}{(1+b)(1+b^2)} \\\\ = \int \left(\frac1{1+b} + \frac{1-b}{(1+b^2)}\right) \, db \\\\ = \frac14 \left(2 \arctan(b) + 2 \ln(1 + b) - \ln(1 + b^2)\right) + C \\\\ = \frac14 \left(2 \arctan(\sqrt a) + 2 \ln(1 + \sqrt a) - \ln(1 + a)\right) + C[/tex]
Noting that a = 0 makes the integral I(a) vanish, we have
[tex]0 = \dfrac14 \left(2 \arctan(\sqrt0) + 2\ln(1 + \sqrt0) - \ln(1 + 0)\right) + C \implies C = 0[/tex]
and so
[tex]\displaystyle I(a) = \frac\pi{4\sqrt2} \left(2 \arctan(\sqrt a) + 2 \ln(1 + \sqrt a) - \ln(1 + a)\right)[/tex]
We recover the integral we want with a = 1, which gives a value of
[tex]\displaystyle \int_0^\infty \frac{\sqrt x \arctan(x)}{1 + x^2} \, dx = \boxed{\frac{\pi^2 + 2\pi\ln(2)}{4\sqrt2}}[/tex]
what is the slope of the line that goes through the points (-3,8) and (0,23)?
Answer:
m=5 or the slope is 5
Step-by-step explanation:
This is because you use the expression y2-y1/x2-x1
A farmer sold 53.2 pounds of carrots and 29.4 pounds of green beans to a restaurant. How many pounds of these two vegetables did the restaurant buy? Explain your thinking. *
Answer:
just add up the 2 numbers pretty simple
Step-by-step explanation:
53.2 plus 29.4
Answer: 82.6 lb
Step-by-step explanation:
Since the restaurant is buying the pounds given from the farmer, add the pounds together.
Add
53.2 + 29.4 = 82.6
Hence, the restaurant bought 82.6 pounds of vegetables.
PLEASE SOLVE ASAP!
The sum of the measures of two complementary angles is equal to ____.
Answer: Two angles are called complementary when their measures add to 90 degrees. Two angles are called supplementary when their measures add up to 180 degrees.
Step-by-step explanation:
Answer:
90°
Step-by-step explanation:
A psychology professor wants to know whether verbal ability is related to memory quality in current first-year students at her small college. Participants in the study (first-year students at her college) complete an online memory task. The students are first shown a list of 60 words. Next they are shown a list of 10 words that were on the original list. Then they are asked to identify the words on the second list that appeared on the original list. She uses the percentage of words that were correctly recognized on the original list as her measure of memory quality. She also asks the students to report several characteristics such as their age, gender, and verbal SAT score. Each of the 750 first-year students (338 males and 412 females) at her school volunteers to participate. The professor chose 75 students at random to complete the memory task and answer the questions. The average percentage of words that were correctly recognized on the original list was 68%. The professor infers that if all 750 first-year students had completed the study, the results would show that an average of 68% (plus or minus sampling error) of the words were correctly recognized as being on the original list. Which of the following are variables in the study? Check all that apply.
A. The students' verbal SAT scores
B. The students' percentage of words that were correctly recognized on the original list
C. The 75 students
D. The 750 students
Answer:
The variable are
A and B
Step-by-step explanation:
Generally we can define a variable as a name of a placeholder representing a value now considering the option to be selected from we see that
The students' verbal SAT scores is a variable because it is a phrase or a place holder that represent a value (in the is case a numerical value) which the score its self
The second option The students' percentage of words that were correctly recognized on the original list is also a variable because it is a placeholder of a name (in this case a phrase ) that represented the actual value itself
The third option The 75 students is not a variable because it is not representing any value but itself is the value
The third option The 750 students is not a variable because it is not representing any value but itself is the value
Consider the functions f(x) = 5x – 2, g(x) = 5x² + 2 and r(x) = –2x + 4.
Evaluate f(g(5)).
10 points!
Answer:
f(-2x+4) = -10x+18
Step-by-step explanation:
Answer:
733 I hope that helps! Sorry if its incorrect
A box contains 3 green marbles, 5 blue marbles, and 7 red marbles. Three marbles are selected at random from the box, one at a time, without replacement. Find the probability that the first two marbles selected are not red, and the last marble is red. Round your answer to four decimal places.A box contains 3 green marbles, 5 blue marbles, and 7 red marbles. Three marbles are selected at random from the box, one at a time, without replacement. Find the probability that the first two marbles selected are not red, and the last marble is red. Round your answer to four decimal places.
Answer:
The probability is [tex]P(K) = \frac{ 28 }{195} [/tex]
Step-by-step explanation:
From the question we are told that
The number of green marbles is [tex]n_g = 3[/tex]
The number of red marbles is [tex]n_b = 5[/tex]
The number of red marbles is [tex]n_r = 7[/tex]
Generally the total number of marbles is mathematically represented as
[tex]n_t = n_r + n_g + n_ b[/tex]
[tex]n_t = 7 + 3 + 5[/tex]
[tex]n_t = 5 [/tex]
Generally total number of marbles that are not red is
[tex]n_k = n_g + n_ b[/tex]
=> [tex]n_k = 3 + 5[/tex]
=> [tex]n_k = 8[/tex]
The probability of the first ball not being red is mathematically represented as
[tex]P(r') = \frac{n_k}{n_t}[/tex]
=> [tex]P(r') = \frac{ 8}{15}[/tex]
The probability of the second ball not being red is mathematically represented as
[tex]P(r'') = \frac{n_k - 1}{n_t -1}[/tex]
=> [tex]P(r'') = \frac{ 8 -1 }{15-1}[/tex] (the subtraction is because the marbles where selected without replacement )
=> [tex]P(r'') = \frac{ 7 }{14}[/tex]
The probability that the first two balls is not red is mathematically represented as
[tex]P(R) = P(r') * P(r'')[/tex]
=> [tex]P(R) = \frac{ 8}{15} * \frac{ 7 }{14}[/tex]
=> [tex]P(R) = \frac{ 8 }{30}[/tex]
The probability of the third ball being red is mathematically represented as
[tex]P(r) = \frac{n_r}{ n_t -2}[/tex] (the subtraction is because the marbles where selected without replacement )
[tex]P(r) = \frac{7}{ 15 -2}[/tex]
=> [tex]P(r) = \frac{7}{ 13}[/tex]
Generally the probability of the first two marble not being red and the third marble being red is mathematically represented as
[tex]P(K) = P(R) * P(r)[/tex]
[tex]P(K) = \frac{ 8 }{30} * \frac{7}{ 13}[/tex]
=> [tex]P(K) = \frac{ 28 }{195} [/tex]
Heeeeeeeeeeeeeeeeeelpppppppppppp
Answer:
As both these angles form a linear pair, they together will form 180°
So,
(2x + 45)° + x = 180°
=> 2x + 45 + x = 180°
If my answer helped, kindly mark me as the brainliest!
Thank You!!
what is the answer to this simultaneous equation
2x+3y=13
4x-y=-2
While vacationing, Rosalie sees a mountain and its reflection on the still water of a lake. She draws the mountain and its reflection as a projection on the coordinate grid, treating the x-axis as the line of reflection. In her drawings, she represents the summit of the mountain using the coordinates (-6,8) .
What are the coordinates of the summit in her projection?
will mark brainlest
Answer:
I would say (2,-9) is her projection
Answer:
ok bet my niqgqa
Step-by-step explanation:
Division using place value step by step
2,440 / 8 I know answer. I just need shown steps please & thank you!
Answer:
see attached
Step-by-step explanation:
The usual methods of long division use place value. Each quotient digit is aligned over the appropriate dividend digit to give it the proper place value.
Hydrogen peroxide decomposes back to water and oxygen when exposed to air and light. When bought at a pharmacy for home use, hydrogen peroxide is sold in dark bottles that are labeled as a concentration of 3% by weight. A chemistry student wants to test the concentration of hydrogen peroxide that has been decanted to light and dark bottles, and exposed to air for 10, 20 and 30 hours. The concentration is tested by titration with potassium permanganate, and each sample is tested twice.answer choices are: color/hours, 2, 6, 12, dark/light, 10/20/30 hours, and concentrationquestions: Response variable (s):Factor(s):Experimental Unit (s):Replication(s):Treatment(s):
Answer:Response variable (s)-concentration, (s): Treatment- 2,6,12 hours,Replication(s)-10/20/30 hours
Experimental Unit- dark/light
Factors-color/hours
Step-by-step explanation:
Factors refer to explanatory variables tat are being studied in an experiment. They are manipulated in order to obtain a response (dependent) variable.
Treatments refer to a combination of factor levels in the experiment. Each factor has two or more levels
The experimental unit refers to the physical entities assigned at random to treatments during the experiment.
Replication refers to the number of times the same experiment is carried out. It enhances the reliability of the result of an experiment.
The concentration is tested by titration with potassium permanganate, and each sample is tested twice Replication-10/20/30 hours.
We have to determine
The concentration is tested by titration with potassium permanganate, and each sample is tested twice.
What is hydrogen peroxide?Hydrogen peroxide is a chemical compound that is gaseous in nature. Its viscosity is greater than that of water.
Since it is unstable it slowly decomposes in the presence of light to form water and oxygen.
The control experiment is necessary for the determination of reliability and statistical significance of results obtained.
Replication helps to ensure accuracy and randomization aids us in making quality generalizations from experimental data.
Testing the concentration of hydrogen peroxide decanted to dark bottles is the control experiment.
Testing the concentration of the species after exposure to air for 10,20 and 30 hours is randomization.
Replication refers to the number of times the same experiment is carried out. It enhances the reliability of the result of an experiment.
Testing each sample twice is replication.
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Write 37/3 as a mixed number
Answer:
12 1/3
Step-by-step explanation:
Answer:
12 1/3
Step-by-step explanation:
1. Divide the numerator by the denominator.
Divide 37 by 3, which equals 12 with a remainder of 1.
37 ÷ 3 = 12 r1
2. Find the whole number.
The whole number is the number of times the denominator divides into the numerator.
3 (the denominator) divides into 37 (the numerator) 12 times, so 12 is the whole number.
Make the remainder the new numerator.
Because the remainder (the leftover value) of dividing 37 by 3 is 12, 1 is the new numerator. The denominator is still 3. So, the mixed fraction is:
12 1/3
Hope it helps...
write the following numbers in words 560 451 2008
Answer: five billion six hundred four million five hundred twelve thousand eight
Step-by-step explanation:
GIVING EXTRA POINTS AND BRAINLIEST
Answer:
1. d<=200
2. f>=35
Step-by-step explanation:
Answer:
d ≤ 200
f ≥ 35
Step-by-step explanation:
"no more than 200 " means it can be equal to or less than 200. Therefore, we would use the "less than or equal to" sign, or "≤". We would put 200 at the end as it is no more than 200 feet, so 200 is the max.
"at least 35" means it is equal to or more than 35. Therefore, we would use the "greater than or equal to sign" or "≥". Since it is talking about 35 miles, we put the 35 at the end with the sign pointing away from it since 35 is the lesser number.
Remember, the "mouth" or opening of the equality sign always points to the greater or bigger number. You can think of it as an alligator wanting to eat the bigger number.
Hope this helps! :)
The sum of two numbers is (−12). What two numbers could you have?
Answer:
you could have something like umm -36 + 30 It also gives the answer as -12 so I guess you could choose any two numbers that add up to -12
Which number is divisible by both 5 and 6?
O A) 132,359
OB) 142,645
OC) 164,780
OD) 193,560
Answer: D/193,560
Step-by-step explanation: 193,560 divided by 5 = 38712
193,560 divided by 6 = 32260
Answer:
D
Step-by-step explanation:
Which is larger, 3.2 x 10^4 or 3.2 x 10^5
Answer:
3.2 x 10^5
Step-by-step explanation:
Answer:
3.2 x 10^5
Step-by-step explanation:
it had a bigger exponent.
!!!!Translate the sentence into an equation!!
The sum of 7 times a number and 3 is 6.
Answer:
7x + 3=6 that is the equation
A 28-story hotel building has 18,000 square feet of space for rooms per floor. The average room is 16 ft x 28 ft. How many rooms do you have available?
Answer:
1,120 rooms
Step-by-step explanation:
You need to find out how many hotel rooms are on one floor, and then multiply that by 28. So each room is 16ft x 28ft, which comes out to be 448ft^2. Then to find how many of these rooms go onto one floor, you must divide 18,000 by 448. This equals about 40. So 40 rooms per floor, times 28 floors is: 1,120. And that's your answer! Make sure you understand how we got that. If you have any questions I can help! :)
Coffee beans sell for $8.00 per pound. How much would 5 1 Over 4
pounds of coffee beans cost?
please help i need this urgently
Answer:
Angle YZX = 19.7 degrees
This is the same as saying angle Z = 19.7 degrees
====================================================
Explanation:
Since there are only three letters, and nothing else, we can refer to "angle YZX" as "angle Z" to shorten things. The middle letter is what determines the vertex of the angle.
With Z as the reference angle, side XY = 7.5 is the opposite side. It's the leg furthest from the reference angle. SIde ZY = 22.3 is the hypotenuse. The hypotenuse is always opposite the 90 degree angle.
We'll use the sine rule since it connects the opposite and hypotenuse
sin(angle) = opposite/hypotenuse
sin(Z) = XY/ZY
sin(Z) = 7.5/22.3
sin(Z) = 0.336323
Z = arcsin(0.336323)
Z = 19.653008983888
Z = 19.7
In this case, rounding to 3 sig figs means rounding to 1 decimal place (the first two digits are part of the whole number while the last digit is the decimal portion).
Make sure your calculator is in degree mode. The notation "arcsin" refers to arcsine which is the same as inverse sine. Your calculator would likely have the button [tex]\sin^{-1}[/tex] to indicate inverse sine.
Answer:
Degrees: 19.7
Radians: 0.343
Step-by-step explanation:
First, let x = angle YZX.
You can solve this using sine (which as a reminder, is opposite over hypotenuse):
sin(x) = 7.5/22.3
x = arcsin(7.5/22.3)
Now you can grab out a calculator and evaluate.
If it's in degrees: 19.7
If it's in radians: 0.343
(Remember to round to 3 significant figures like the problem said)
I hope this helped you.
Question
Solve the following system of equations.
- x² – 3y² = - 12
–4x² - y² = -37
Provide your answer below:
I
I need the exact answer please. There are 8 sets of answers for this question what are they fill in the question mark. I hope I’ll get my answers quickly. Thank you
(?,?),(?,?),(?,?),(?,?)
Answer:
(3,1) (3,-1) (-3,1) (-3,-1)
Step-by-step explanation:
use the model to solve 4 friends equally split in one half of a pound of candy how much will each friend get
Answer:
.25 pounds
Step-by-step explanation:
Unfortunately, the type of model that is supposed to be used is not given so I will just solve what I can. So each friend gets and equal amount. If we have 1 pound in the beginning, and 4 friends, then we have to divide the candy by 4. So we do 1 pound/4=.25 pounds. If you want to go above and beyond, we can always convert the pounds into ounces and make it so each friend gets 4 ounces of candy.
An animal shelter has a ratio of dogs to cats that's 3:2. If there are 45 animals at the shelter, how many
dogs are there?
A. 15
B. 90
C. 20
D. 45
PLZ SOLVE FAST FOR 800 ROBUX TOO
Answer:
[tex] x = 22.5[/tex]
[tex]180 - 4x + 2x = 6x \\ x = 180 \div 8 = 22.5 \\ so \: x = 22.5[/tex]
Answer:
22.5
Step-by-step explanation:
username Maxracker1
2 2/5 x 5/7 in simplest form
Answer:
1 5/7
Step-by-step explanation:
2 2/5 x 5/7
= 12/5 x 5/7
= 12/7
1 5/7
What is the slope of the line passing through the points (-2, 6) and (1, -3)
Answer:
slope is -3
Step-by-step explanation:
use slope equation, shown in picture hope this helps
Slope worksheet. Please help me!!!
Find the unit rate. 3/8 mile in 3/4 hour
We are required to find the unit rate, that is, miles per hour
The unit rate is 1/2 miles per hourNumber of miles = 3/8 mileNumber of hours = 3/4 hourTherefore, unit rate = = 3 / 8 ÷ 3/4= 3 / 8 × 4/3= 12/24= 1/2Read more:
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3 times what equals -18
Answer:
-6
Step-by-step explanation:
Let "what" be termed as "x".
Set the equation:
3 * x = -18
Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Divide 3 from both sides:
(3x)/3 = (-18)/3
x = -18/3
x = -6*
*Note: When you divide a negative number with a positive number, your answer will be negative.
-6 is your answer.
~
The value of number which is 3 times to -18 is -6 .
Given,
3 *x = -18
So,
Let "what" be termed as "x".
Form the equation including the variable x ,
3 * x = -18
Isolate the variable, x.
Note the equal sign, what you do to one side, you do to the other. Divide 3 from both sides:
(3x)/3 = (-18)/3
x = -18/3
x = -6
-6 is the required answer .
Know more about equations,
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