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The value of int(0)^(1)(x^(4)(1-x)^(4))/...

The value of `int_(0)^(1)(x^(4)(1-x)^(4))/(1+x^(4))dx` is (are)

A

`(22)/(7)-pi`

B

`(2)/(105)`

C

0

D

`(71)/(15)-(3pi)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `I=int_(0)^(1)(x^(4)(1-x)^(4))/(1+x^(2))dx = int_(0)^(1)((x^(4)-1)(1-x)^(4)+(1-x)^(4))/((1+x^(2)))dx`
`=int_(0)^(1)(x^(2)-1)(1-x)^(4)dx+int_(0)^(1)((1+x^(2)-2x)^(2))/((1+x^(2)))dx`
`=int_(0)^(1){(x^(2)-1)(1-x)^(4)+(1+x^(2))-4x+(4x^(2))/((1+x^(2)))}dx`
`=int_(0)^(1){(x^(2)-1)(1-x)^(4)+(1+x^(2))-4x+4-(4)/(1+x^(2)))dx`
`=int_(0)^(1)(x^(6)-4x^(5)+5x^(4)-4x^(2)+4-(4)/(1+x^(2)))dx`
`=[(x^(7))/(7)-(4x^(6))/(6)+(5x^(5))/(5)-(4x^(3))/(3)+4x-4 tan^(-1)x]_(0)^(1)`
`=(1)/(7)-(4)/(6)+(5)/(5)-(4)/(3)+4-4((pi)/(4)-0)=(22)/(7)-pi`
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