∫ (x³cos(x/2)+(1/2))√(4-x²) dx [-2,2]
Distribute
∫ x³√(4-x²)cos(x/2) + (1/2)√(4-x²) dx [-2,2]
Now
Determine the function of
f(x) = x³√(4-x²)cos(x/2)
Is an odd or a function
f(-x) = -x³√(4-x²)cos(x/2)
Therefore the function is odd so automatically the answer for
∫ x³√(4-x²)cos(x/2) dx [-2,2] = 0
While for
(1/2)∫ √(4-x²) dx[-2,2]
f(-x) = √(4-x²)
So the function is even therefore
(1/2)•2 ∫ √(4-x²) dx [0,2]
∫ √(4-x²) dx [0,2]
x = 2sinθ
dx = 2cosθ dθ
2 ∫ √(4-4sin²θ)cosθ dθ
4 ∫ cos²θ dθ
2 ∫ 1+cos(2θ) dθ
2(θ+(1/2)sin(2θ))|[0,2]
2θ+sin(2θ)|[0,2]
2arcsin(x/2)+(1/2)x√(4-x²)|[0,2]
2arcsin(1)+0
= 2(π/2)
= π
Us humans have a purpose that many haven't found out yet. Maybe I'm wrong so it's okay to judge me for saying this..
We have the brains to think that none other living things in this world can do. We can think of things that is helpful for the world we're walking. We can maintain outgrowing populations of other species that if it become outnumbered it will be unstable for the world. We can think for ways balance the ecosystem we have the minds that can think of ways to help other living creatures like we can maintain a species that is close to its extinction. The only problem with us humans that we have been sinned. 7 sins to be exact that thinking we are the alphas because we have the wits that other animals don't have. We can think of helping trees to grow more and naturally faster. We can think of ways to protect other wildlife. But the only problem with having the way of thinking unlike nothing else is the pride that kept our minds that we are the alpha predators. We cut trees and kill animals to feed our satisfaction and it's getting out of hand despite of we can use our minds to help others. We can kill animals but only to balance their population. I hope many of us can still think of passing back a thanks to nature like maintain the population of animals and planting trees
