The first integral has a well-known beta function representation, so the second one should too. The beta function itself is defined as
[tex]B(x,y) = \displaystyle \int_0^1 t^{x-1} (1-t)^{y-1} \, dt[/tex]
and satisfies the identity
[tex]\displaystyle B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}[/tex]
Later on, we'll also use the so-called reflection formula for the gamma function; for non-integer z,
[tex]\Gamma(z) \Gamma(1-z) = \dfrac{\pi}{\sin(\pi z)}[/tex]
as well as the identity
[tex]\dfrac{\Gamma(z+1)}{\Gamma(z)} = z[/tex]
Replace [tex]x\to\sin^{-1}(x)[/tex] in both integrals, so that
[tex]\displaystyle \int_0^{\frac\pi2} \sqrt{\sin(x)} \, dx = \int_0^1 \frac{\sqrt x}{\sqrt{1-x^2}} \, dx[/tex]
[tex]\displaystyle \int_0^{\frac\pi2} \frac{dx}{\sqrt{\sin(x)}} = \int_0^1 \frac{dx}{\sqrt x \sqrt{1-x^2}}[/tex]
Now replace [tex]x\to\sqrt x[/tex] :
[tex]\displaystyle \int_0^1 \frac{\sqrt x}{\sqrt{1-x^2}} \, dx = \frac12 \int_0^1 x^{-\frac14} (1-x)^{-\frac12} \, dx = \frac12 B\left(\frac34, \frac12\right) [/tex]
[tex]\displaystyle \int_0^1 \frac{dx}{\sqrt x \sqrt{1-x^2}} = \frac12 \int_0^1 x^{-\frac34} (1-x)^{-\frac12} \, dx = \frac12 B\left(\frac14, \frac12\right)[/tex]
So, the original integral (which I condense here to a double integral) is
[tex]\displaystyle \int_0^{\frac\pi2} \int_0^{\frac\pi2} \sqrt{\frac{\sin(x)}{\sin(y)}} \, dx \, dy = \frac14 B\left(\frac34, \frac12\right) B\left(\frac14, \frac12\right)[/tex]
[tex]\displaystyle = \frac14 \frac{\Gamma\left(\frac14\right) \Gamma\left(\frac34\right) \Gamma\left(\frac12\right)^2}{\Gamma\left(\frac54\right) \Gamma\left(\frac34\right)}[/tex]
[tex]\displaystyle = \frac14 \frac{\Gamma\left(\frac14\right) \Gamma\left(\frac34\right) \Gamma\left(\frac12\right)^2}{\frac14 \Gamma\left(\frac14\right) \Gamma\left(\frac34\right)}[/tex]
[tex]\displaystyle = \Gamma\left(\frac12\right)^2 = \frac{\pi}{\sin\left(\frac\pi2\right)} = \boxed{\pi}[/tex]
An apartment complex is turning the grassy area next to the current parking lot into more parking space.Once they are done,how much total parking area will there be?
Once the apartment complex have been turned to a parking area, the total parking area will be 1620 square units
How to determine the total parking area?From the complete question, the parking area consists of the following shapes and dimensions
Rectangle: 28 units by 40 unitsRectangle: 20 units by 25 unitsThe area of a rectangle is the product of its dimensions.
So, the area of the parking area is:
Area = 28x40 + 20x25
Evaluate the products
Area = 1120 + 500
Evaluate the sum
Area = 1620
Hence, the total area of the parking area is 1620 square units
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Question 5
O Mark this question
22
5.3
A manufacturing company has two plants at different locations producing three different
items equally within each plant. Based on the number of workers and the demand for
the items in their respective locations, the number of items manufactured per day by
each plant is listed in the table.
5.4 T
5.4
5.4
Plant A
Plant B
5.4
Item 1
22
15
5.4
Item 2
00
8
12
5.4.
Item 3
14
25
5.5 An
5.5.1
Select the observed and expected frequencies for Item 2 produced by Plant B.
5.5.2
5.5.3
Observed: 12
5.5.4
The observed and expected frequencies for Item 2 produced by Plant B will be 12 and 17.3.
What is an observed frequency?It should be noted that observed frequency simply means counts that are made from experimental data.
Here, the observed frequencies for Item 2 produced by Plant B is 12. The expected frequency will be:
= (15 + 12 + 25)/3
= 52/3
= 17.3
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Hewo Pls Answer the questions below TY!
Answer:
dont worry if people report you for a answer that is not right lol trust me i know
Step-by-step explanation:
find the area of the shaded region. Use the formula A= pie r^2 to find the area of a circle
Answer:
(π6x+21π)m²
Step-by-step explanation:
Solution,
For small circle
radius(r)=x+2
Area(A1)=πr²
Area(A1)=π×(x+2)²
Area(A1)=π×(x²+4x+4)
Area(A1)=πx²+π4x+4π
For big circle
radius(r)=x+5
Area(A2)=πr²
Area(A2)=π×(x+5)²
Area(A2)=π×(x²+10x+25)
Area(A2)=πx²+π10x+25π
Again,
Area of shaded part=A2-A1
Area of shaded part=πx²+π10x+25π-(πx²+π4x+4π)
Area of shaded part=πx²+π10x+25π-πx²-π4x-4π
Area of shaded part=π6x+21π
The function f is continuous on the interval (4, 15), with some of its values given in the table above.(in the picture)
Estimate the average value of the function with a Left Rectangle
Approximation, using the 4 intervals between those given points.
(I will vote and like, thank you)
Answer:
-11.545
Step-by-step explanation:
1/11(5*-6 + 2*-11 + 3*-18 + 1*-21)
1. All of the following are ways to show information on a map except--
Question 1 options:
Symbols
Themes
Colors
Contours
Question 2 (1 point)
Saved
The Black Forest of Germany and forest located in China and America suffer from–
Question 2 options:
Deforestation
Drought
Acid Rain
Blight
Question 3 (1 point)
Saved
Which source would provide the most accurate information about a group’s culture?
Question 3 options:
Satellite Images
Elevation Maps
World Globe
Field work
Question 4 (1 point)
Saved
During the time of Christopher Columbus, which continent would have been most accurately mapped?
Question 4 options:
Europe
Asia
South America
North America
Question 5 (1 point)
Two bodies of water greatly affected by irrigation are--
Question 5 options:
Baltic Sea and Volga River
Aral Sea and Colorado River
Black Sea and Thames River
Caspian Sea and Seine River
Question 6 (1 point)
What do mountainous regions do to create flat land for farming?
Question 6 options:
Polders
Terracing
Canals
Deforestation
Question 7 (1 point)
When put together what does a line of latitude and longitude create?
Question 7 options:
Absolute Location
Relative Location
Point
Pole
Question 8 (1 point)
This process brings water to crops.
Question 8 options:
Irrigation
Polders
Terracing
Erosion
Question 9 (1 point)
Temperature, precipitation, and seasons are all ______________________.
Question 9 options:
Latitude Zones
Vegetation Characteristics
Climate Characteristics
Seasons
Question 10 (1 point)
Tropical/hot climate found between the Tropic of Cancer and the Tropic of Capricorn
Question 10 options:
Polar Latitude Climate Zone
High Latitude Climate Zone
Middle Latitude Climate Zone
Low Latitude Climate Zone
1. The ways to show information on a map include using symbols, colors, and contours, except B. Themes.
What is the information shown on a map?A map displays information about distance and direction, using symbols, colors, and contours.
Typically, maps are drawn to carefully reduced scales.
Some maps display the following information:
Political boundariesPopulationPhysical featuresNatural resourcesRoadsClimatesElevation (topography)Economic activities.2. The Black Forest of Germany and forests located in China and America suffer from C. Acid Rain.
What is acid rain?Acid rain refers to sulphuric or nitric acid, falling to the ground from air, in the form of fog, hail, and dust.
Originating from atmospheric pollution, acid rain is known to cause environmental harm to:
TreesFreshwaters and soilsAquatic life-formsPaintings (peeling)Steel structures such as bridgesStone buildings and sculpturesHuman life.3. The source that provides the most accurate information about a group's culture is D. Fieldwork.
What is fieldwork?Fieldwork is research conducted in a natural environment for obtaining primary data.
Fieldwork gives the most accurate information about a group's culture better than Satellite Images, Elevation Maps, and the World Globe because it is based on the group's space.
4. During the time of Christopher Columbus, the continent that would have been most accurately mapped was A. Europe.
What is mapping?Mapping refers to the creation of maps.
For instance, Europe was most accurately mapped than Asia, South America, and North America during the time of Columbus.
5. The two bodies of water greatly affected by irrigation are B. the Aral Sea and the Colorado River.
What is irrigation?Irrigation is the application of water to soil to meet the needs of growing plants and ensure a bountiful harvest for farmers.
Types of irrigation systems include:
Sprinkler irrigationSurface irrigationDrip irrigationSub-irrigationManual irrigation.6. Mountainous regions create flat land for farming by B. Terracing.
What is terracing?Terracing is the method of growing crops on the sides of hills or mountains by constructing specific ridged platforms (or graduated terraces).
Terracing decreases erosion and surface runoff, supporting the growth of crops requiring irrigation, such as rice.
7. When put together, a line of latitude and longitude creates A. Absolute location.
What is an absolute location?An absolute location refers to a precise point on Earth or another defined space.
8. The process that brings water to crops is A. Irrigation, as previously defined.
9. Temperature, precipitation, and seasons are all C. Climate Characteristics.
What are climate characteristics?Climate characteristics refer to climate features such as windiness, humidity, cloud cover, atmospheric pressure, and fogginess
10. Tropical/hot climate found between the Tropic of Cancer and the Tropic of Capricorn is referred to as the D. Low Latitude Climate Zone.
What does the low latitude climate zone imply?The low latitude climate zone implies a tropical climate with warm temperatures.
Thus, the low latitude zone covers the areas of the earth south of the equator and north of about 30 degrees latitude.
Learn more maps and climate zones at https://brainly.com/question/23439088, https://brainly.com/question/18641405, https://brainly.com/question/23391170
PLEASE HELP PLEASE PLEASE
Answer:
C but if its not correct pls tell me. :)
Step-by-step explanation:
Answer:
D
Step-by-step explanation:
The formula for volume is length × width × height
2×1×2= 4
3×1×1= 3
4×1×2= 8
3×1×3= 9
So D has the biggest volume.
A student generates two patterns.
• Pattern P starts at 8 and follows the rule "multiply by 2, subtract 3" to get the
next term.
• Pattern Q starts at 6 and follows the rule “subtract 2, multiply by 2" to get the
next term.
Which lists show the first five terms of each pattern?
I) Pattern P: 8, 10, 14, 22, 38
Pattern Q: 6, 8, 12, 20, 36
1) Pattern P: 8, 13, 23, 43, 83
Pattern Q: 6, 10, 18, 34, 66
1.) Pattern P: 8, 16,5, 32, 2
Pattern Q: 6, 4, 12, 2, 24
1) Pattern P: 8, 13, 23, 43, 83
Pattern Q: 6, 8, 12, 20, 36
Answer:
pattern P: 8,13,23,43,83
pattern Q: 6,10,18,34,66
The tables that contain the first 5 elements of both patterns are:
Pattern P: 8, 13, 23, 43, 83Pattern Q: 6, 8, 12, 20, 36How to find the first 5 terms of each pattern?For pattern P we start at 8, and to get the next element we multiply by 2, and then subtract 3.
So the first 5 elements are:
[tex]a_1 = 8\\\\a_2 = 2*8 - 3 = 13\\\\a_3 = 2*13 - 3 = 23\\\\a_4 = 2*23 - 3 = 43\\\\a_5 = 2*43 - 3 = 83[/tex]
For pattern Q we start at 6, and then we subtract 2 and multiply by 2 to get the next element.
[tex]b_1 = 6\\\\b_2 = (6 - 2)*2 = 8\\\\b_3 = (8 - 2)*2 = 12\\\\b_4 = (12 - 2)*2 = 20\\\\b_5 = (20 - 2)*2 = 36\\\\[/tex]
Then the tables that contain the first 5 elements of both patterns are:
Pattern P: 8, 13, 23, 43, 83Pattern Q: 6, 8, 12, 20, 36If you want to learn more about patterns:
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Razonamiento matemático
Maria tiene $900 pesos y Juan tiene el doble de la tercera parte de María.
Si la suma del dinero de Pedro y Juan es de $1000 pesos ¿cuánto dinero tiene Pedro?
Resolviendo un par de ecuaciones lineales veremos que pedro tiene 400 pesos.
¿Cuánto dinero tiene Pedro?Sabemos que:
María tiene 900 pesos.
Juan tiene el doble de la tercera parte de María, entonces:
J = (2/3)*$900 = $600
Juan tiene 600 pesos.
Finalmente, sabemos que la suma del dinero de Pedro y Juan es de $1,000 pesos, entonces tenemos que resolver la ecuación lineal:
P + J = $1,000
P + $600 = $1,000
P = $1,000 - $600 = $400
Pedro tiene 400 pesos.
Sí quieres aprender más sobre ecuaciones lineales, puedes leer:
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Easy math help picture below 2 problems
Answer: 10 Inches
Step-by-step explanation:
Different events have the following likelihoods.sort them from least to greatest:
[tex]\frac{1}{x^2-5x}=\frac{x+7}{x}-1[/tex]
Answer:
x=5.14285
Step-by-step explanation:
give brainliest pls
Answer:
Refer to the attachment.
________
Hope it helps ⚜
Evaluate the expression
If x=5
24x + 16
what the denominator has been multiple by
[tex] \frac{.1}9 = \frac{?}{27} [/tex]
Answer:
it has been multiplied by 3
Step-by-step explanation:
9 x 3=27
1 x 3=3
Please help in urgent need for this answer
Step-by-step explanation:
i)
Consider the exponential equation
[tex]y = a {e}^{bt} [/tex]
Now Substitute these points to get a and b value
(0,37000) , (1,20350)
[tex]37000 = a \times {e}^{b \times 0} [/tex]
[tex]37000 = a \times 1[/tex]
Hence a=37000
Now Substitute t=1
[tex]20350 = 37000 \times {e}^{b \times 1} [/tex]
[tex] \frac{20350}{37000} = {e}^{b} [/tex]
[tex]0.55 = {e}^{b} [/tex]
[tex]b = log_{e}(0.55) = - 0.5978[/tex]
b≈(-0.6)
[tex]y = 37000 {e}^{ - 0.6t} [/tex]
is the required equation
ii)
[tex]y = 37000 {e}^{ - 0.6 \times 0.4} [/tex]
[tex]y = 37000 {e}^{ - 0.24} [/tex]
iii) it decreases by a factor of
[tex] {e}^{0.6} [/tex]
every decade
So the factor by which the price of car decreases every year is
[tex] {e}^{ \frac{0.6}{10} } [/tex]
as 1year =1/10 decade
Hence the required factor
[tex] {e}^{0.06} [/tex]
I hope it helped you
Sofia can text 40 words in one minute
At that rate, how many words can she
text in 12 minutes?
From 32 + 14i, subtract its complex conjugate. What is the difference?
Answer:
To find the complex conjugate, negate the term with i
.
32−14i
Step-by-step explanation:
CAN SOME TELL ME THE ANSWER
Answer:
54 kids per bus
Step-by-step explanation:
first take 7-331 and you get 324 divided by 6= 54
is this a funcation? (5,8) (7,0) (0,5) (5,2) (9,1)
What is line r called and what are ∠1 and ∠2?
The line r is the transversal and ∠1 and ∠2 are linear pairs.
Angles formed when a transversal cuts parallel linesWhen a transversal cuts parallel lines , angles formed includes corresponding angles, alternates angles, linear pair angles etc.
Therefore, p and q are the parallel lines.
The transversal is r.
The transversal r, cuts the parallel lines p and q.
Therefore, line r is the transversal and ∠1 and ∠2 are linear pairs.
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Answer:
D. Line r is the transversal and ∠1 and ∠2 are consecutive interior angles
Step-by-step explanation:
took the quiz
Evaluate the function rule for the given value.
y = 8 ∙ 3x for x = 4
1944
647
96
648
Answer:
96
Step-by-step explanation:
y=8.3x
x=4
Substituting x as 4,
y=8*3*4
y=8*12
y=96
Hence, x=4, y=96
What is the value of min the figure below?
11
m
с
O A. 336
O B. 77
O C. V178
D. V624
O E. 42
F. 126
[tex] {m}^{2} = 7 \times (7 + 11)[/tex]
[tex] {m}^{2} = 7 \times 18[/tex]
[tex] {m}^{2} = 126[/tex]
[tex]m = \sqrt{126} [/tex]
prove the equation, you can only change one side
csc0-sin0=cot0cos0
[tex]\textit{Pythagorean Identities} \\\\ sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ csc(\theta )-sin(\theta )~~ = ~~cot(\theta )cos(\theta ) \\\\[-0.35em] ~\dotfill\\\\ csc(\theta )-sin(\theta )\implies \cfrac{1}{sin(\theta )}-sin(\theta )\implies \cfrac{1~~ - ~~sin^2(\theta )}{sin(\theta )} \\\\\\ \cfrac{cos^2(\theta )}{sin(\theta )}\implies \cfrac{cos(\theta )}{sin(\theta )}\cdot cos(\theta )\implies cot(\theta )cos(\theta )[/tex]
what is the digit 9 in this number 195,487
Answer:
90,000 the answer to that question
Answer:
The ten thousands place.
What is the length of the line segment with endpoints (-6.4) and (2,-5)
Answer:
I guess the answer is √145
please help me with this its due in 15 minutes plus ill need help with more
Answer:
171 Square Feet
Step-by-step explanation:
We first cut the figure up into smaller parts...
we then use the formula (L x W = A) to find the area
L = Lenth
W = Width
the first rectangle (1)...
15 x 5 = 75
the second figure (2)...
8 x 9 = 72
the third figure (3)...
6 x 4 = 24
now we have to add up all the values to find the area of the entire figure...
75 + 72 + 24 = 171
Hope this helps! Please let me know if you need more help or think my answer is incorrect. Brainliest would be MUCH appreciated. Also remember to rate answers to help other students, While leaving thanks to answers that help you. Have a great day!
The population of a country can be calculated by the function P(t) = 3.32 t5/3. P represents the population (in millions) and t is time (in years). Find the population after 3 years.
Terrance draws a triangle with sides of 10 cm, 11 cm, and 12 cm with angle measures of 55°, 60°, and 65°. Which of the following is a possible way to classify the triangle by side and angle measurements?
A. Obtuse and isosceles
B. Acute and scalene
C. Obtuse and equilateral
D. Acute and equilateral
Answer:
Acute and Scalene
Step-by-step explanation:
The measurments of the triangle are all almost equal and all the angles are less than 90 degrees, so all of them are acute angles. The scalene triangle has almost all equal sides.
The possible way to classify the triangle by side and angle measurements is,
⇒ Acute and scalene.
What is mean by Angle?An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.
We have to given that;
Terrance draws a triangle with sides of 10 cm, 11 cm, and 12 cm with angle measures of 55°, 60°, and 65°.
Since, All the angles are less than 90 degree.
Hence, It shows the Acute angle.
Here, Terrance draws a triangle with sides of 10 cm, 11 cm, and 12 cm.
Clearly, All the sides of triangle are not same in length.
Thus, It shows scalene.
Thus, The possible way to classify the triangle by side and angle measurements is,
⇒ Acute and scalene.
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Use the table and a calculator to help you find a solution to 0.1 • 2 x = , rounded to the nearest whole number.
x 0.1 • 2 x
1 0.2 4
2 0.4 2
3 0.8 1.33
4 1.6 1
5 3.2 0.8
A solution to this equation to the nearest whole number is x = .
Does this equation have any other solutions?
The solution of the equation is approximately x = 4 and there are no other solutions
How to determine the solution of the equation?The equation is given as
0.1(2^x) = 4/x
Also, we have the table of values as one of the given parameters
Next, we check the table of values
From the table of values, we can see that:
The value of x is between x = 3 and x = 4
This is so because the values of the expressions 0.1(2^x) and 4/x are closest at these values
However, the value of x is closer to 4 than 3
This means that
x = 4 (approximately)
Hence, the solution of the equation is approximately x = 4 and there are no other solutions
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Choose a natural number between 1 and 100, inclusive. What is the probability that the number chosen is not a multiple of 4? (Enter the probability as a fraction.)
Answer: 3/4
Step-by-step explanation:
Every fourth number is divisible by 4 and every 3 out of 4 is not divisible.
There are 25 multiples of 4 (4*1 through 4*25)
So the indicated probability = 75/100
= 3/4