Let `y=f(x)=4x^(3)+2x-6`, then the value of `int_(0)^(2)f(x)dx+int_(0)^(30)f^(-1)(y)dy` is equal to _________.

Answer

56

Explanation

`f(1)=0` and `f(2)=30`
`:.int_(0)^(2)f(x)dx+int_(0)^(30)f^(-1)(y)dy`
`=int_(0)^(1)f(x)dx+int_(1)^(2)f(x)dx+int_(0)^(30)f^(-1)(y)dy`
`=int_(0)^(1)(4x^(3)+2x-6)dx+int_(1)^(2)f(x)dx+int_(1)^(2)tf'(t)dt`
`=[x^(4)+x^(2)-6x]_(0)^(1)+int_(1)^(2)f(x)dx+[tf(t)]_(1)^(2)-int_(1)^(2)f(t)dt`
`=-4+2f(2)-f(1)`
`=-42xx30-f(1)`
`=56`