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
Suppose a family drives at an average rate of 60 mi/h on their way to visit relatives and then an average rate of 40 mi/h on the
hammer [34]

Answer:

Part A) 2 hours

Part B) d/60 hours

Part C) 48 miles/hour

Step-by-step explanation:

Part A) Let "d" be the distance in miles the family traveled to visit their relatives. How many hours did it take to drive there?

we know that

Let

x ----> the time it take on their way to visit relatives in hours

y ---> the time it take on the way back in hours

we know that

The distance of the way to visit is equal to the distance of the way back

The speed multiplied by the time is equal to the distance

so

60x=40y -----> equation A

y=x+1 -----> equation B

substitute equation B in equation A and solve for x

60x=40(x+1)

60x=40x+40

20x=40

x=2 hours

Find the value of y

y=2+1=3 hours

d=60x

The time it take on their way to visit relatives is 2 hours

The time it take on the way back is 3 hours

Part B) In terms of "d," how many hours did it take to drive there?

we have that

The distance is equal to the speed multiplied by the time

d=60x

Solve for x

x=d/60 hours

Part C) Write and solve an equation to determine the distance the family drove to see their relatives. What was the average rate for the entire trip?

To find the average rate for the whole trip we will need the total distance and the total time

The distance of the way to visit is

d=60x

substitute the value of x

d=60(2)=120 miles

so

The total distance is

120(2)=240 miles

we know that the total time is

2 + 3 =5 hours

The average rate for the complete trip is

240/5=48 miles/hour

5 0
3 years ago
Read 2 more answers
1. What is the distance between the points (-3,0) and (3, 8) in the xy-plane?
olga_2 [115]
The answer is c
Ok please thanks me if I’m correct
If is not correct comment
8 0
3 years ago
3250-1250=1500+2(1250-500) is this true
exis [7]

Answer:

Its false because

Step-by-step explanation:

3250-1250=1500+2.(1250-500)

2000 = 1500 + 2 . 750

2000 = 1500 + 1500

2000 != 3000

Obviously 2000 isnt the same of 3000

5 0
3 years ago
Read 2 more answers
PLz FAAAAAAAAASTTTTTTTTT!!!!!!!! HAAALPP
bonufazy [111]

Answer:

any value of a makes the equation true.

Step-by-step explanation:


4 0
3 years ago
Determine if the two figures are congruent and explain your answer using transformations.
Maslowich

The coordinate A(-1, 1) reflected over y-axis is (1, 1) is the coordinate of E, hence the two figures are congruent

  • The given figures are quadrilaterals, in order to determine whether they are similar, we need to check if they are reflections of each other.

  • For the Quadrilateral ABCD, the coordinate of A is at A(-1, 1)  and for the Quadrilateral DEFG, the coordinate of E is at E(1, 1).

  • Note that if an object is reflected over the y-axis the transformation is (x, y)->(-x, y)

  • We need to check whether if we reflect the coordinate A over the y-axis we will get coordinate E

Since the coordinate A(-1, 1) reflected over y-axis is (1, 1) is the coordinate of E, hence the two figures are congruent

Learn more on reflections here:brainly.com/question/1908648

7 0
2 years ago
Other questions:
  • Find the distance between the points (4/4,1/10) and (2/5,7/10)​
    15·1 answer
  • State the value of the discriminant. Then determine the number of real roots of the equation.
    11·2 answers
  • 7 hours equals how many seconds
    6·2 answers
  • (WILL MARK BRAINLEST!)
    11·1 answer
  • The perimeter of a standard sized rug is 40 feet. The width is 2 feet less than the length. Find the dimensions.
    6·1 answer
  • Determining the End Behavior of a Polynomial Function HELP ASAP !!!!
    15·1 answer
  • What is the sum of the solutions of the 2 equations? 4x = 12 & 2y + 10 = 22
    15·1 answer
  • The length of a football field should be measured in
    6·2 answers
  • -7x+6y=34 7x+4y=-24 helppp
    8·1 answer
  • Choose the equation that is equivalent to: 4(8x+6)=20
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!