How Do You Solve This Complex Integral Involving Integration by Parts?

[johnq2k7]

Evaluate the integral below:

2Pi times the integral of (2-y^2)(sqrt(1+4y^2)) dy from 0 to sqrt(2)

work shown:

using integration by parts I got:

let u= sqrt(1+4y^2)

therefore du= 4y/(4y^2+1)

let dv= (2-y^2)

therefore V= 2y- y^3/3

since Integration by Parts is (u)(V) - integral of (V)(du)

therefore i got 2*Pi times (sqrt(1+4y^2)(2y-y^3/3) - integral of (2y-y^3/3)(4y/sqrt(4y^2+1)) dy from 0 to sqrt (2)

after evaluating this i keep getting a more completed integration by parts.. and continuous iteration.. please help me solve this integral

 

[Dr.D]

Looks like you messed up du.

 

[johnq2k7]

Looks like you messed up du.

i believe my du is correct, even so... i believe my integration by parts method is tedious and it won't result in a solution.. i believe I'm using the wrong method to integrate this equation... please help