[tex] \rm \int_{0}^ \infty \frac{ \sqrt[ \scriptsize\phi]{x} \tan^{- 1} (x)}{(1 + {x}^{ \phi} {)}^{2} } {}^{} {} \: dx\\ [/tex]​

Answers

Answer 1

With ϕ ≈ 1.61803 the golden ratio, we have 1/ϕ = ϕ - 1, so that

[tex]I = \displaystyle \int_0^\infty \frac{\sqrt[\phi]{x} \tan^{-1}(x)}{(1+x^\phi)^2} \, dx = \int_0^\infty \frac{x^{\phi-1} \tan^{-1}(x)}{x (1+x^\phi)^2} \, dx[/tex]

Replace [tex]x \to x^{\frac1\phi} = x^{\phi-1}[/tex] :

[tex]I = \displaystyle \frac1\phi \int_0^\infty \frac{\tan^{-1}(x^{\phi-1})}{(1+x)^2} \, dx[/tex]

Split the integral at x = 1. For the integral over [1, ∞), substitute [tex]x \to \frac1x[/tex] :

[tex]\displaystyle \int_1^\infty \frac{\tan^{-1}(x^{\phi-1})}{(1+x)^2} \, dx = \int_0^1 \frac{\tan^{-1}(x^{1-\phi})}{\left(1+\frac1x\right)^2} \frac{dx}{x^2} = \int_0^1 \frac{\pi2 - \tan^{-1}(x^{\phi-1})}{(1+x)^2} \, dx[/tex]

The integrals involving tan⁻¹ disappear, and we're left with

[tex]I = \displaystyle \frac\pi{2\phi} \int_0^1 \frac{dx}{(1+x)^2} = \boxed{\frac\pi{4\phi}}[/tex]


Related Questions

Adrianna has $20 in her savings account. Each week she will add $5 to it. Which equation represents this situation?

Answers

y is total amount, x is weeks

we would have
y = 5x + 20

Answer:

Let y = the amount in Adrianna's savings account

Let x = the number of weeks passed

y = 5x + 20

Simplify cos^4x+2cos^2xsin^2x+sin^4x

Answers

Answer: 1

Step-by-step explanation:

Compared with the graph of the parent function,
which equation shows a vertical stretch by a factor
of 6, a shift of 7 units right, and a reflection over the
X-axis?

Answers

Step-by-step explanation:

as I don't see any equations or graphs here I can only answer in general :

f(x) is the patent function.

g(x) is the transformed function.

g(x) = -6f(x-7)

the "-" reflects the curve over the x-axis (what was positive is now negative and vice versa).

the factor 6 stretches the curve up and down by the factor 6 (everything is now 6 times larger or smaller).

and the argument "x-7" relates x (for the new function) with "x-7", meaning that now things are happening at x, that were originally happening at x-7, so they happen now "later" (more to the right on the x-axis) than before - hence a shift to the right.

Answer:

its D

Step-by-step explanation:

took the test :)

Please help solve this SAT math problem. I have 4 more like these so need help with this so that I can solve the rest by myself. Thank you.

Answers

The notation just means

(f/g)(-4) = f(-4) / g(-4)

If f(x) = x² + 2 and g(x) = 1 - 3x, then

f(-4) = (-4)² + 2 = 16 + 2 = 18

g(-4) = 1 - 3 (-4) = 1 + 12 = 13

and so

(f/g)(-4) = 18/13

Melanie goes to a shelter to buy a dog. She has settled on two dogs, an Alsatian and a bulldog. To decide between them, she plans to flip a coin. Before she does, she flips the coin 50 times to check for fairness of the model. Her results are shown in the table.

Coin Toss Result Times Result Occurred
heads 24
tails 26
The relative frequency of landing on heads is ____. The relative frequency of landing on tails is
____.

Answers

Using it's concept, it is found that the relative frequency of landing on heads is 0.48 and of landing on tails is 0.52.

What is a relative frequency?

A relative frequency is given by the number of desired outcomes divided by the number of total outcomes.

In this problem, out of 50 flips, 24 landed on heads and 26 on tails, hence:

Fh = 24/50 = 0.48.

Ft = 26/50 = 0.52.

The relative frequency of landing on heads is 0.48. The relative frequency of landing on tails is 0.52.

More can be learned about relative frequency at https://brainly.com/question/23359601

Given that a= 6 cm and b = 5 cm, work out the area of the triangle.

Answers

Answer:

15cm²

Step-by-step explanation:

Area of a triangle: 1/2*b*h

a = height (6cm)

b = base (5cm)

A = 1/2* b * h

A = 1/2 * 5 * 6

A = 1/2 * 30

A = 15cm² <== final answer

Hope this helps!

A suit is on sale for 72% off. The sale price is $263.20.
What is the regular price?

Answers

Answer:

940

Step-by-step explanation:

To find the original price of something that been discount do the following:

Subtract the discount from 100 to get the percentage of the original price.Multiply the final price by 100.Divide by the percentage in Step One.

Or we can do the following:

We shall consider the original price as x; hence the discount can be considered as (x−263.20). We can write the equation:

x · 72/100=x - 263.20

x · 18/25 = x - 263.20

Multiply both sides by 25

18x = 25(x - 263.20)

Open the brackets.

18x = 25x-6580

Add 6580 to each side.

6580 +18x = 25x

Subtract 18x from each side.

6580 = 7x

Subtract 7x from each side.

x = 940

Check Answer:

940 * 72% = 676.80

940 - 676.80=263.20

Hence, the regular price is 940.

~lenvy~

Step-by-step explanation:

The sale price ( The marked price ) :- $263.20

Discount :- 72%

Discount :- marked 100 ,Discount 72 ,Seliing ( Regular ) 28

Regular price :- 263.20 × 28 / 100

$ 73.69

solve this sum urgent no spam​

Answers

Answer:

Step-by-step explanation:

YES IS THE ANSWER BECAUSE X=9

AND 11+3=87

1st is commutative property of addition

2nd is commutative property of multiplication

If a coat was marked down 20% and the sale price is $55.60, what is the original price?

Answers

Answer:

$69.50

Step-by-step explanation:

As it was marked down 20%, this means the sale price would be 100-20=80%.  

When                   80%  ---- cost $55.60      

Therefore when 100% ---- cost = $55.60 * 100/80

So when we solve this, we get the final ans $69.50                                  

Please solve with explanation

Answers

Try this option, all the details are in the attachment. Note the equation a) is performed with green, the equation b) with red line.

With break green line is the cureve y= -x²; with break red line is y=2x².

What is the equation of the axis of symmetry?

Answers

Answer: x=7
Explanation: graph :)

A shipping company will ship a package for $7.50 when the volume is no more than 15,000 cubic centimeters. Grace needs to ship a package that is 3x-5 centimeters long, 2x centimeters wide, and x + 20 centimeters tall. A.Write a polynomial expression to represent the situation if Grace plan as to spend a maximum of $7.50
B. Write and solve a system of equations
C. What should the dimensions of the package be to have the maximum volume?

Answers

The maximum volume is at x = 0.85 cm. The inequality is 6x³ + 110x² - 200x ≤ 15000

What is an inequality?

An inequality is an expression that shows the non equal comparison between two or more numbers and variables.

A shipping company will ship a package for $7.50 when the volume is no more than 15,000 cubic centimeters. Hence, if Grace plan as to spend a maximum of $7.50:

Volume (V) = (3x - 5)(2x)(x + 20) = 6x³ + 110x² - 200x

The inequality is:

6x³ + 110x² - 200x ≤ 15000

6x³ + 110x² - 200x - 15000 ≤ 0

x = 10

The maximum volume is at:

V' = 0

V' = 18x² + 220x - 200 = 0

x = 0.85 cm

The maximum volume is at x = 0.85 cm. The inequality is 6x³ + 110x² - 200x ≤ 15000

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A pizza is divided into 12 slices. If there are eight slices left, what fraction of the pizza is
remaining?

Answers

Answer:

when 12 pieces say =1/12

so 8 pieces say = 1/8

Step-by-step explanation:

Answer:

2/3

Step-by-step explanation:

fraction of remaining part=8/12 (divided with 4)

=2/3

The Adventure Zone amusement park asked all of their employees who attend college how many hours of
homework they had per week and determined that they had an average of 24 hours of homework per
week.
Based on this data, which of the following conclusions are valid?
Choose 1 answer:
24% of all of the amusement park employees who attend school have homework each week.
24% of all of the amusement park employees who attend college have homework each week.
The amusement park employees who attend college have an average of 24 hours of
homework per week, but we cannot conclude anything about the percent of the population
who have homework.

Answers

The amusement park employees who attend college have an average of 24 hours of homework per week, but we cannot conclude anything about the percent of the population who have homework.

What is the valid conclusion?

The survey was to determine the average number of hours per week for employees that attended college. The result was that college students had an average of 24 hours of homework per week.

To learn more about surveys, please check: https://brainly.com/question/18521835

Answer:

the last

answer

Step-by-step explanation:

Which expression is equivalent to the following complex fraction?

Answers

Answer:

Option B

Step-by-step explanation:

Numerator:

[tex]\dfrac{3}{x -1 }-4 =\dfrac{3}{x-1}-\dfrac{4*(x-1)}{x-1}\\\\[/tex]

                 [tex]=\dfrac{3- 4x -4*(-1)}{x-1}=\dfrac{3-4x+4}{x-1}\\\\\\=\dfrac{-4x+7}{x-1} \: \textbf{ [Combine like terms]}[/tex]

Denominator:

[tex]2-\dfrac{2}{x-1}=\dfrac{2*(x-1)}{x-1}-\dfrac{2}{x-1}\\[/tex]

                [tex]= \bf \dfrac{2x-2-2}{x-1}\\\\\\= \dfrac{2x-4}{x-1}\\\\= \dfrac{2*(x- 2)}{x-1}[/tex]

[tex]\dfrac{\dfrac{3}{x-1}-4}{2-\dfrac{2}{x-1}}=\dfrac{\dfrac{-4x+7}{x-1}}{\dfrac{2*(x-2)}{x-1}}[/tex]

                 [tex]\bf = \dfrac{-4x+7}{x-1}*\dfrac{x-1}{2(x-2)} \ \ \ \textbf{ [(x-1) in the numerator and denominator will be cancelled]}\\\\\\=\dfrac{-4x+7}{2(x-2)}[/tex]

find the equation (in terms of x) of the line through the points (-4,2) and (3,-5)

Y=

Answers

Answer:   y = -x-2

==================================================

Explanation:

First we need the slope

[tex](x_1,y_1) = (-4,2) \text{ and } (x_2,y_2) = (3,-5)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{-5 - 2}{3 - (-4)}\\\\m = \frac{-5 - 2}{3 + 4}\\\\m = \frac{-7}{7}\\\\m = -1\\\\[/tex]

Then we can find the y intercept using the coordinates of (x,y) = (-4,2)

y = mx+b

2 = -1(-4)+b

2 = 4+b

2-4 = b

-2 = b

b = -2

Alternatively, you can use the other point (x,y) = (3,-5) and m = -1.

Because m = -1 and b = -2, we go from y = mx+b to y = -1x+(-2) which simplifies to y = -x-2

mathematics pls help 4 questions will give brain to best answer

Answers

Answer:

1)ur first answer is 12

2)Your median is 9

3)9

4) range 7

Step-by-step explanation:

it’s the most common number

The middle numbers are 8 and 10.9 is the number in between them which mean it is the median

4) you subtract the biggest number with the smallest one 12-5=7

Answer:

[1] 12

[2] 9

[3]  9

[4] 5

Step-by-step explanation:

[1] What is the mode for the following set of data?

5,7,8,10,12,12

Answer = 12

Explanation: Mode is what most appears in data set. Thus, we see that 12 is the mode.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[2] What is the median for the following set of data?

5,7,8,10,12,12

Answer = 9

Explanation:

Median is basically what in the middle.

5,7,8,10,12,12

5,7,8,10,12,12

Since there are two number in the middle: Add them up and then divide by 2. 8 + 10 = 18/2 = 9

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

[3] What is the mean for the following set of data?

5,7,8,10,12,12

Answer = 9

Explanation:

Mean is add all the number up and then divide by total number in data set

5 + 7 + 8 + 10 + 12 + 12 = 54

Total number in data set : 6

54/6 = 9

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

[4] What is the range for the following set of data?

5,7,8,10,12,12

Answer = 5

Explanation:

Range = IQR = Q3 - Q1

Knowing that;

Quartiles:

Q1 --> 7

Q2 --> 9

Q3 --> 12

Hence, 12 - 7 =5

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Lenvy~

write balanced chemical equation for the process of photosynthesis giving the physical state of all the substances involved in the conditions of the reaction​

Answers

Answer:

The process of photosynthesis is commonly written as: 6CO2 + 6H2O → C6H12O6 + 6O2.

Step-by-step explanation:

PLEASE ANSWER FAST WILL MARK BRAINLIEST

Answers

[tex]\sf y=50\left(1.8\right)^{x}[/tex]

=================================

the first term : 50

crosses point : (0, 50), (1, 90)

solve for y,

[tex]\sf y=50\left(1.8\right)^{1}[/tex]

[tex]\sf y=90[/tex]

Thus D is correct option.

Graph:

What is the distance
between (-6, -2) and (2, 4)?

Answers

Answer:

10

Step-by-step explanation:

distance = √(-6-2)^2+(-2-4)^2

Ans=10

[tex]\qquad \qquad \huge \pink {\sf{☁Answer☁}} \\ \\ [/tex]

[tex] \rule{60mm}{2.2pt}[/tex]

[tex] \large \purple{ \rm{Given ↦}} \\ [/tex]

A ( -6 , -2 ) ( x1 , y1 )B ( 2 , 4 ) ( x2 , y2 )

[tex] \rule{60mm}{2.2pt}[/tex]

[tex] \large \purple{ \rm{By \: DISTANCE \: FORMULA ↦}} \\ \\ \rm{AB =\sqrt{( x2-x1)^{2} +(y2-y1) ^{2} }}[/tex]

[tex] \rule{60mm}{2.2pt}[/tex]

Substituting values

[tex] \bold {AB= \sqrt{( - 6 - 2 )^{2} + ( - 2 - 4) ^{2}} } \\ \\ \bold {AB= \sqrt{( - 8) ^{2} + ( - 6) ^{2} }} \\ \\ \bold {AB= \sqrt {(64) + (36)} } \\ \\ \bold {AB= \sqrt{(100)}} \\ \\ \bold {AB= 10 \: units.}[/tex]

[tex]\purple{\rule{15mm}{2.9pt}} \red{\rule18mm{2.5pt}} \orange{ \rule18mm{2.5pt}}[/tex]

[tex]\sf{\:мѕнαcкεя\: ♪...}[/tex]

A library had 6422 music CDs stored on 26 shelves.If the same number of CDs were stored on each shelf how many CDs were stored on each shelf

Answers

one shell stores CDs:6422/26

246

Subtract 55 from the sum of 234 and 8

Answers

234+ 8= 242
242-55=187
187 is the answer :)

please help
The radius of the base of a right circular cone is 5 times greater than the radius of a second right circular cone. If the heights of both cones are the same, what is the volume of the larger cone divided by the volume of the smaller cone?
A. 5
B. 10
C. 15
D. 25

Answers

Answer: its A or B I think its more of B than A

Find the value of the expression.

Answers

Answer:

0

Step-by-step explanation:

((3/5 ^0)^-2)

3/5^0=0

0^-2= 0

What is the value of x? 1/6 (3 - 2x) < 1/3 x

Answers

Step-by-step explanation:

1/6 (3 - 2x) < 1/3 x

let's multiply everything by 6

3 - 2x < 6/3 x

3 - 2x < 2x

3 < 4x

3/4 < x

so, x > 3/4

since 3/4 is never equal to x but is purely smaller than x, the interval of valid x values starts right after 3/4 and goes to infinity.

x = 3/4 is not valid, because 3/4 is not smaller than 3/4 (it is equal to) ...

4sin(-x)cos(-x)= -2sin2x
true or false

Answers

[tex]\text{L.H.S}\\\\=4\sin(-x) \cos(-x)\\\\=-4\sin x \cos x\\\\=-2 \cdot 2 \sin x \cos x\\\\=-2 \sin 2x\\\\=\text{R.H.S}\\\\\text{The identity is true.}[/tex]

[tex] \rm \int_{0}^2 \left( \sqrt[3]{ {x}^{2} + 2x} + \sqrt{1 + {x}^3 } \right )dx \\ [/tex]​

Answers

If f(x) has an inverse on [a, b], then integrating by parts (take u = f(x) and dv = dx), we can show

[tex]\displaystyle \int_a^b f(x) \, dx = b f(b) - a f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) \, dx[/tex]

Let [tex]f(x) = \sqrt{1 + x^3}[/tex]. Compute the inverse:

[tex]f\left(f^{-1}(x)\right) = \sqrt{1 + f^{-1}(x)^3} = x \implies f^{-1}(x) = \sqrt[3]{x^2-1}[/tex]

and we immediately notice that [tex]f^{-1}(x+1)=\sqrt[3]{x^2+2x}[/tex].

So, we can write the given integral as

[tex]\displaystyle \int_0^2 f^{-1}(x+1) + f(x) \, dx[/tex]

Splitting up terms and replacing [tex]x \to x-1[/tex] in the first integral, we get

[tex]\displaystyle \int_1^3 f^{-1}(x) \, dx + \int_0^2 f(x) \, dx = 2 f(2) - 0 f(0) = 2\times3+0\times1=\boxed{6}[/tex]

Melanie invested $63,000 in an account paying an interest rate of 7\tfrac{1}{8}7
8
1

% compounded quarterly. Lillian invested $63,000 in an account paying an interest rate of 7\tfrac{3}{8}7
8
3

% compounded continuously. After 15 years, how much more money would Lillian have in her account than Melanie, to the nearest dollar?

Answers

Go study at your house don’t come again

john had thrice as many stickers as Ken. After Ken had given away 12 stickers to his brother, John had 5 times as many stickers as Ken. How many stickers did John and Ken have in the end?

Answers

The number of stickers John and Ken have in the end is 6 and 18 respectively.

How to write and solve equation?

let

Ken stickers = xJohn stickers =3x

New stickers

Ken = x - 12John = 5x

x + x - 12 = 3x + 5x

2x - 12 = 8x

-12 = 8x - 6x

-12 = -2x

x = 12/2

x = 6

So,

Ken stickers = x

= 6 stickers

John stickers = 3x

= 3 × 6

= 12 stickers

Learn more about equation:

https://brainly.com/question/1214333

Helppppp!!! 50 pts!

Translate the sentence into an equation.

Three times the sum of a number and 5 is 7.

Use the variable C for the unknown number.

Answers

Answer:

3(C + 5) = 7

Step-by-step explanation:

Let C = unknown number

The sum of a number and 5:

C + 5

Three times the sum of a number and 5:

3(C + 5)

Three times the sum of a number and 5 is 7:

3(C + 5) = 7

To solve

Use the distributive law:  a(b + c) = ab + ac

⇒ 3C + 15 = 7

Subtract 15 from both sides:

⇒ 3C = -8

Divide both sides by 3:

⇒ C = -8/3

Le the number be c

Sum of a number and c means c+53times that means 3(c+5)

So

The expression is

3(c+5)=7

On solving

3c+15=73c=-8c=-8/3
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