The value of `int_(1)^(3)(sqrt(1+(x-1)^(3))+(x^(2)-1)^(1/3)+1)dx` is __________.

Answer

8

Explanation

`I=int_(1)^(3)sqrt(1+(x-1)^(3))dx+int_(1)^(3)((x^(2)-1)^(1/3)+1)dx`
`=int_(1)^(3)f(x)dx+int_(1)^(3)f^(-1)(x)dx`
Insecond integral put `f^(-1)(x)=y`
`:.x=f(y)`
`:. dx=f'(y)`
Also if `1=sqrt(1+(x-1)^(3)),x=1`
and if `3=sqrt(1+(x-1)^(3)),x=3`
`:.I=int_(1)^(3)f(x)dx+int_(1)^(3)yf'(y)dy`
`=int_(1)^(3)f(x)dx+[yf(y)]_(1)^(3)-int_(1)^(3)f(y)dy`
`=3xxf(3)1xxf(1)`
`-9-1`
`=8``