Expert: Socrates - 3/13/2006

Question
evaluate

S from 0 to 1 x times the square root of 1-x4

not sure how to write this on a computer?

s=integral sign

x = variable

1-x is all under the square root sign and only the x is to the 4th power
maybe this will help evaluate the integral from 0 to 1 x(1-x to the 4th power) the ( ) represent the square root sign

Answer
You want S x(1 - x^4)^1/2 dx from 0 to 1

First, make the substitution y = x^2
Then dy/dx = 2x and dx = dy/2x

the integrand x(1 - x^4)^1/2 dx becomes

x(1-y^2)^1/2 dy/2x = (1/2)(1-y^2)^1/2 dy

The limits for new the integral in y are found using y = x^2,
when x=0 , we get y=0 , and when x=1, we get y=1. So the limits for the integral in the new variable y are the same as before.

We are still left with the problem of integrating
(1/2)(1-y^2)^1/2 dy from 0 to 1

Make another substitution, this time let y = cosz

Then dy/dz = -sinz and dy = -sinz dz

The integrand now becomes

(1/2)(1-(sinz)^2)^1/2 (-sinz) dz

Since 1-(sinz)^2 = (cosz)^2 ,

the integrand becomes

(1/2)(cosz)(-sinz) dz

To find the new limits for the integral in the variable z , use y=cosz and z = cos^-1 y .

When y=0, we have z = pi/2 . When y=1 , we find z=0

So, we now need to find

(-1/2) S (sinz)(cosz) dz with limits from pi/2 to 0

This can be rewritten as

(1/2) S (sinz)(cosz) dz with limits from 0 to pi/2

since sinz cosz = (1/2) sin2z , this integral is equal to


(1/4) S sin2z dz

An anti derivative for sin2z is (-1/2)cos2z ,

so we evaluate cos2z at z = pi/2 , z = 0 , find the difference , then multiply by (1/4)(-1/2) = -1/8

cos(2(pi/2)) = cos pi = -1

cos(2(0)) = cos 0 = 1


the difference is -1 - 1 = -2

multiply by -1/8 and get (-1/8)(-2) = 1/4

The answer is 1/4