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](https://tex.z-dn.net/?f=I%20%3D%20%5Cdisplaystyle%20%5Cint_0%5E%5Cinfty%20%5Cfrac%7B%5Csqrt%5B%5Cphi%5D%7Bx%7D%20%5Ctan%5E%7B-1%7D%28x%29%7D%7B%281%2Bx%5E%5Cphi%29%5E2%7D%20%5C%2C%20dx%20%3D%20%5Cint_0%5E%5Cinfty%20%5Cfrac%7Bx%5E%7B%5Cphi-1%7D%20%5Ctan%5E%7B-1%7D%28x%29%7D%7Bx%20%281%2Bx%5E%5Cphi%29%5E2%7D%20%5C%2C%20dx)
Replace 
:
Split the integral at x = 1. For the integral over [1, ∞), substitute 
:
The integrals involving tan⁻¹ disappear, and we're left with
Answer:
Step-by-step explanation: the first is m the second one is 15x+5
Answer:
By the Empirical Rule, 99.7% of the students have grade point averages that are between 1.28 and 3.8.
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 2.54
Standard deviation = 0.42.
Between 1.28 and 3.8?
1.28 = 2.54 - 3*0.42
So 1.28 is 3 standard deviations below the mean
3.8 = 2.54 + 3*0.42
So 3.8 is 3 standard deviations above the mean
By the Empirical Rule, 99.7% of the students have grade point averages that are between 1.28 and 3.8.
Answer:
Considering is a compunded increase rate of the yearly car insurance fee and x is the initial value of the insurance fee then
x×(1.05)^12=$86.82
×=$86.82/(1.05)^12
Answer:
the base of the ramp is 31 inches long.
Step-by-step explanation:
c = a/sin(α)
= 33.48514
b = √c2 - a2
= √33.4851373155042 - 122
= √977.25442103816
= 31.26107
