1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
tekilochka [14]
2 years ago
5

5D%7Bx%7D%20%20%5Ctan%5E%7B-%201%7D%20%28x%29%7D%7B%281%20%2B%20%20%7Bx%7D%5E%7B%20%5Cphi%7D%20%7B%29%7D%5E%7B2%7D%20%20%7D%20%7B%7D%5E%7B%7D%20%20%20%7B%7D%20%20%20%5C%3A%20dx%5C%5C%20" id="TexFormula1" title=" \rm \int_{0}^ \infty \frac{ \sqrt[ \scriptsize\phi]{x} \tan^{- 1} (x)}{(1 + {x}^{ \phi} {)}^{2} } {}^{} {} \: dx\\ " alt=" \rm \int_{0}^ \infty \frac{ \sqrt[ \scriptsize\phi]{x} \tan^{- 1} (x)}{(1 + {x}^{ \phi} {)}^{2} } {}^{} {} \: dx\\ " align="absmiddle" class="latex-formula">​
Mathematics
1 answer:
Rasek [7]2 years ago
8 0

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

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

Replace x \to x^{\frac1\phi} = x^{\phi-1} :

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

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

\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

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

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

You might be interested in
Keith as 11/12 hours to play, he has already played 1/4. which is the best estimate of the fractional part of an hour he has lef
____ [38]

Answer:

The best estimate is greater than 1/2 but less than 3/4

Step-by-step explanation:

If you multiply 1/4 by 3/3 (which is also 1), you can change the denominator without changing the value.  So 1/4 is equal to 3/12.

Since keith has 11/12 hours to play, and he has already played 3/12 hours, subtract 3/12 from 11/12 to get 8/12 hours. This is how much time he has left to play.  

If you simplify 8/12 hours, you get 2/3 hours.

So the best estimate would be: greater than 1/2 but less than 3/4.

4 0
3 years ago
Need answer PLS will give brainiest
Oxana [17]

Answer:

n=-6

Step-by-step explanation:

4 0
3 years ago
What is x cubed in standard form
Gelneren [198K]

Answer:

x^3

Step-by-step explanation:

It stays the same.

Hope this helps.

3 0
3 years ago
Money from an allowance or job is known as what
sineoko [7]

Answer:

salary my guesss don't know right or wrong

3 0
3 years ago
Read 2 more answers
A 20% off coupon for
algol [13]

Answer:

$20

Step-by-step explanation:

<em>20% is equal to 20/100 therefore, if 5 x 20 is 100, you would multiply 5 x 4 and get 20, </em>

the reason you don't subtract the $4 is because it says "was" so the answer is $20

6 0
2 years ago
Other questions:
  • Which of the following volumes represent a perfect cube? Check all that apply.
    13·2 answers
  • A type of elevator has a maximum weight capacity Y1 , which is normally distributed with mean 5000 pounds and standard deviation
    11·1 answer
  • What is the answer for number one and possibly number two please
    6·1 answer
  • the length of one board is 2 feet 10 inches,and the length if anither board is 1 yard 1 foot 9 inches. what is the total length
    9·2 answers
  • This is for algebra 1
    13·1 answer
  • What is four times a number, minus 8, is equal to three times the number, plus 9
    14·1 answer
  • A group of students recorded how much money they earned per week for allowance to the nearest dollar. The students created the f
    7·2 answers
  • Which of the sails on the boats below has a greater area?
    5·1 answer
  • Find the perimeter of the polygon with the given vertices. C(4, -1), D(4,−5), E(2,−3)
    11·1 answer
  • I need help answering this question!! Why does the equation (x-4)^2 – 28 = 8 have two solutions?
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!