If the value definite integral `underset(-4)overset(4)(f)(x^(3)-2x^(2)-"cosec"^(3)x+tanx)/(5-|x|)dx=a+bln5,"where "a,binI`, find the value of `((a+b)/(2))`.

Explanation

As, `(x^(3)-"cosec"^(3)x+tanx)/(5-|x|)` is an odd function.
So, `I=underset(-4)overset(4)(f)(-2x^(2))/(5-|x|)dxrArrI=-4underset(0)overset(4)(f)(x^(2))/(5-|x|)dx`
`I=-4[underset(0)overset(4)(f)(x^(2)-25+25)/(5-x)dx]`
`I=-4[underset(0)overset(4)(f)-(x+5)dx+25underset(0)overset(4)(f)(dx)/(5-x)]`
`I=-4[(-x^(2))/(2)underset(0)overset(4)(|)-(5x)underset(0)overset(4)(|)-25(ln|5-x|)underset(0)overset(4)|]`
`I=-4[(-16)/(2)-20-25(0-ln5)]`
`I=-4[-28+25ln5]rArrI=112-100ln5-=(a+bln5)` (Given)
`rArr" "((a+b)/(2))=((112-100) /(2))=(12)/(2)=6`.