The value of `int_0^1 4x^3 {d^2/(dx^2) (1 - x^2)^5 } dx` is

Answer

2

Explanation

`int_(0)^(1)4x^(3){(d^(2))/(dx^(2))(1-x^(2))^(5)}dx`
`=[4x^(3)d/(dx)(1-x^(2))^(5)]_(0)^(1)-int_(0)^(1)12x^(2)d/(dx)(1-x^(2))^(5)dx`
Integrating using by parts)
`=[4x^(3)xx5(1-x^(2))^(4)(-2x)]_(0)^(1)`
`-12[[x^(2)(1-x^(2))^(5)]_(0)^(1)-int_(0)^(1)2x(1-x^(2))^(5)dx]`
`=0-0-12[0-0]+12int_(0)^(1)2x(1-x^(2))^(5)dx`
`=12[-((1-x^(2))^(6))/6]_(0)^(1)=12[0+1/6]=2`