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The value of int0^1 4x^3{(d^2)/(dx^2)(1...

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

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The correct Answer is:
2
PLAN Integration by parts
`intf(x)g(x)dx = f (x) int g (x) dx - int((d)/(dx)[f(x)] intg (x) dx ) dx`
Given , `I= 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)12 x^(2) (d)/(dx) (1- x^(2))^(5)dx`
`=[4x^(3)xx5(1-x^(2))^(4)(-2x)]_(0)^(1)`
`-12 [ [x^(2)(1-x^(2))^(5)]_(0)^(1)- int 2x(1-x^(2))^(5)dx]`
`=0-0- 12 (0-0)+12 int_(0)^(1)2x (1- x^(2))^(5)dx`
`=12xx[-((1-x^(2))^(6))/(6)]_(0)^(1)=12[0+ (1)/(6)]=2`
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