ज्ञात कीजिए : `int_(0)^(2) (dx)/(( 1 7 + 8 x - 4 x^(2))[e^(6(1-x))+1])`

Answer

`= - (1)/(4 sqrt(21)) [ log |(sqrt(2 1 ) - 2)/(2 + sqrt(21))|]`

Explanation

`I=int_0^2dx/[(17+8x-4x^2)" "`
`" "xx(e^(6(1-x))+1)`

`[because int_0^af(x)dx`

`" "= int_0^af(a-x)dx]`

`=int_0^a1/(17+8x-4x^2)dx`

`=-1/4int_0^2dx/(x^2-2x-17//4)`

`-1/4int_0^2dx/((x-1)^2-21//4)`

`=-1/(4sqrt21)log|(2x-2-sqrt21)/(2x-2+sqrt21)|_0^2`

` rArrI=-1/(8sqrt21)[log|(2-sqrt21)/(2+sqrt21)|`

`" "-log|(2+sqrt21)/(sqrt21-2)|]`

`=-1/(4sqrt21)[log|(sqrt21-2)/(2+sqrt21)|]`