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Science & Mathematics

Do you think that extraterrestrials exist?
Best answer: I promise you they do. They are already here, and there is based on this Earth homie. The moon has maybe over 250 million beings on there. The government is hiding everything, Fr. 
Religious people Did you know we're made of stardust left over from the formation of the sun?
What you may not know is that stars will die and be reborn in the same spot multiple times. This builds up dust and gas, which gravity combines and condenses into other things. It'll take hydrogen and oxygen, two of the most common elements in the universe, and make water, an extremely common compound. Water is the medium.on which our kind of life is based. We are literally made of stardust. you are physically a part of the universe. In you, the universe is aware of itself. 
How much damage would be done if a meteor the size of Russia hit Earth?

How come the universe has not collapsed in a black hole right after the Big Bang?

Is vaping smoking?

Is it space that is expanding or Galaxies?

Is evolution real or not?

Did a comet or asteroid wipe out a worldwide civilization 11,500 years ago?

How many stars in the universe?

What color does it make when you mix a rainbow?

Do you think the majority of humans will live on other planets one day?
Best answer: no other planet in our solar system is inhabitable as we know it 
Would u accept metric system in USA. Meters, kilograms, celsius, liters.?

Math problem(multiplying)?
Can someone please help me with this one 0.0045x2000 I have no idea how to do this. And if you could explain how to do it please. Thank you 
How to identify series and parallel resistors in complex circuit? I am confused.?
For eg:some say same current flows through circuit,then it is series combination of resistors. But I am not getting which resistors are in series and which resistors are in parallel in the complex ciruit. So please help me to identify series and parallel resistors in complex circuits. 
What weighs more? A pound of feathers or a pound of stones?

Need a help in // INTEGRATION // problem....?
Best answer: Note to poster:
This is a fun but infuriating integral. I made several attempts (double integrals, differentiating under the integral sign) on this the night before to no success. One of my attempts was very similar to John's solution, but I too made some error that I could not find. This should finally do it.

First of all, we can rewrite this integral as a double integral:
∫(x = 0 to 1) (ln(1+x²)/(1+x)) dx
= ∫(x = 0 to 1) [x²/(1+x²y)]/(1+x) {for y = 0 to 1} dx
= ∫(x = 0 to 1) ∫(y = 0 to 1) [x² / ((1+x²y) (1+x))] dy dx.
Now, we reverse the order of integration:
∫(y = 0 to 1) ∫(x = 0 to 1) [x² / ((1+x²y) (1+x))] dx dy
= ∫(y = 0 to 1) ∫(x = 0 to 1) (1/(1+y)) * [(x1)/(1+x²y) + 1/(1+x)] dx dy, by partial fractions
= ∫(y = 0 to 1) (1/(1+y)) * [∫(x = 0 to 1) (x/(1+x²y)  1/(1+x²y) + 1/(1+x)) dx] dy
= ∫(y = 0 to 1) (1/(1+y)) [(1/(2y)) ln(1+x²y)  (1/√y) arctan(x√y) + ln(1+x)) {for x = 0 to 1}] dy
= ∫(y = 0 to 1) (1/(1+y)) [(1/(2y)) ln(1+y)  (1/√y) arctan(√y) + ln 2] dy.
Distributing:
∫(y = 0 to 1) [(1/(2y(1+y))) ln(1+y)  (1/√y) arctan(√y)/(1+y) + ln(2)/(1+y)] dy.
By partial fractions, 1/(y(y+1)) = 1/y  1/(y+1).
So, we obtain
∫(y = 0 to 1) [(ln(1+y)/(2y)  ln(1+y)/(2(1+y)))  (1/√y) arctan(√y)/(1+y) + ln(2)/(1+y)] dy.
Integrate each term:
(i) ∫(y = 0 to 1) ln(1+y)/(2(1+y))
= ∫(u = 0 to ln 2) (1/2) u du, letting u = ln(1+y) and du = dy/(y+1)
= (1/4)(ln²2).
(ii) ∫(y = 0 to 1) (1/√y) arctan(√y) dy/(1+y)
= ∫(u = 0 to 1) 2 arctan u du/(1+u²), letting u = √y ==> y = u², dy = 2u du
= ∫(w = 0 to π/4) 2w dw, letting w = arctan u, dw = du/(1+u²).
= π²/16.
(iii) ∫(y = 0 to 1) ln(2) dy/(1+y)
= ln(2) ln(1+y) {for y = 0 to 1}
= ln²2
(iv) [The tough one...]
∫(y = 0 to 1) ln(1+y) dy/(2y)
= (1/2) ∫(y = 0 to 1) ln(1+y) dy/y
= (1/2) ∫(y = 0 to 1) [Σ(n = 1 to ∞) (1)ⁿ⁺¹ yⁿ/n] dy/y, via power series for ln(1+y)
= (1/2) ∫(y = 0 to 1) [Σ(n = 1 to ∞) (1)ⁿ⁺¹ yⁿ⁻¹/n] dy
= (1/2) Σ(n = 1 to ∞) (1)ⁿ⁺¹ yⁿ/n² {for y = 0 to 1}
= (1/2) Σ(n = 1 to ∞) (1)ⁿ⁺¹/n²
= (1/2) (π²/12); see below
= π²/24.
For the next to last step above, note that
Σ(n = 1 to ∞) (1)ⁿ⁺¹/n²
= 1  1/2² + 1/3²  1/4² + 1/5²  1/6² + ...
= (1 + 1/2² + 1/3² + 1/4² + 1/5² + 1/6² + ...)  2(1/2² + 1/4² + 1/6² + ...)
= (1 + 1/2² + 1/3² + ...)  2 * (1/2²)(1 + 1/2² + 1/3² + ...)], by factoring
= (1  1/2)(1 + 1/2² + 1/3² + ...)
= (1/2)(π²/6), since Σ(n = 1 to ∞) 1/n² = π²/6
= π²/12.

Putting this all together,
∫(x = 0 to 1) (ln(1+x²)/(1+x)) dx
= ∫(y = 0 to 1) [(ln(1+y)/(2y)  ln(1+y)/(2(1+y)))  (1/√y) arctan(√y)/(1+y) + ln(2)/(1+y)] dy
= π²/24  (1/4)(ln²2)  π²/16 + ln²2
= (3/4)(ln²2)  π²/48.

I hope this helps! 
Does it take too much time to reach the center of the universe?
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Consider the universe expanded from a single point to where it is now. So, when wasn't EVERYTHING at the center? 
Integrate this linear factor using partial frations?
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Why don't airplanes fly straight into space since the earth is round?