Integrate x√(2-x) ?

Question

thanks

Answer 1

eleventy 2?

Answer 2

You need to understand how to integrate a product of functions using the method of substitution.

Let u=2-x

then express (2-x) and x in terms of u, that is :

(2-u)* Square root of u

expand the bracket and then express the result back in terms of x.

Answer 3

put 2 - x = u²

x = 2 -u ²

dx = -2u du

Answer 4

∫ x √(2 - x) dx =

let √(2 - x) = u →

2 - x = u² →

- x = u² - 2 →

x = 2 - u² →

dx = - 2u du

thus, substituting, you get:

∫ x √(2 - x) dx = ∫ (2 - u²) (u) (- 2u du) =

∫ - (u² - 2) (- 2u²) du =

2 ∫ (u² - 2) u² du =

expand it into:

2 ∫ (u^4 - 2u^2) du =

2 ∫ u^4 du - 2 ∫ 2u^2 du =

2 ∫ u^4 du - 4 ∫ u^2 du =

2 [u^(4+1)]/(4+1) - 4 [u^(2+1)]/(2+1) + C =

(2/5) u^5 - (4/3) u^3 + C

finally, substituting back u = √(2 - x), you get:  

∫ x √(2 - x) dx = (2/5) √(2 - x)^5 - (4/3) √(2 - x)^3 + C

I hope it helps...

Bye!