Evaluate the following: `int_(-1)^(1)(sin x-x^(2))/(3-|x|)dx`

Answer

`7-18log_(e)(2//3)`

Explanation

`int_(-1)^(1)(sinx-x^(2))/(3-|x|)dx`
`=int_(-1)^(1)(sinx)/(3-|x|)dx-int_(-1)^(1)(x^(2))/(3-|x|)`
`=0-2int_(0)^(1)(x^(2))/(3-|x|)dx` [ `:' (sinx)/(3-|x|)` is odd and `(x^(2))/(3-|x|)` is even ]
`=-2int_(0)^(1)(x^(2)dx)/(3-|x|)=2int_(0)^(1)(x^(2))/(x-3)dx`
`=2int_(0)^(1)(x+3+9/(x-3))dx`
`=2[(x^(2))/2+3x+9log|x-3|]_(0)^(1)`
`=[7+18"log"2/3]`