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Evaluate the integrals int0 2(dx)/(x+4-x...

Evaluate the integrals `int0 2(dx)/(x+4-x^2)`

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`int_(0)^(2) (1)/(x+4-x^(2))dx`
`=int_(0)^(2)(1)/(x+4-x^(2)-((1)/(4))^(2)+((1)/(2))^(4))dx`
`=int_(0)^(2) (1)/(4+(1)/(4)-[(x^(2)-x+(1)/(4))])dx`
`=int_(0)^(2)(1)/(((sqrt(17))/(2))^(2)-(x-(1)/(2))^(2))dx`
`=(1)/(2.(sqrt(17))/(2))[log |((sqrt(17))/(2)+x -(1)/(2))/((sqrt(17))/(2)-x+(1)/(2))|]_(0)^(2)`
`=(1)/(sqrt(17))[log |(sqrt(17)+2x-1)/(sqrt(17)-2x+1)|]_(0)^(2)`
`=(1)/(sqrt(17))[log|(sqrt(17)+3)/(sqrt(17)-3)|-log |(sqrt(17)-1)/(sqrt(17)+1)|]`
`=(1)/(sqrt(17))log |(sqrt(17)+3)/(sqrt(17)-3)+(sqrt(17)-1)/(sqrt(17)+1)|`
`=(1)/(sqrt(17))log |((sqrt(17)+3)(sqrt(17)+1))/((sqrt(17)-3)(sqrt(17)-1))|`
`=(1)/(sqrt(17))log |(20+4sqrt(17))/(20-4sqrt(17))|`
`=(1)/(sqrt(17))log |((5+sqrt(17))^(2))/((5)^(2)-sqrt(17))xx(5+sqrt(17))/(5+sqrt(17))|`
`[ :' (a-b)(a+b)=a^(2)-b^(2)]`
`=(1)/(sqrt(17))log |(25+17+10sqrt(17))/(25-17)|`

`=(1)/(sqrt(17))log |(42+10sqrt(17))/(8)|`
`=(1)/(sqrt(17))log |(21+5sqrt(17))/(4)|`
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