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Question
Evaluate the integral by using substitution.
`int_0^2 dx/(x + 4 - x^2)`
Solution
Let `I = int_0^2 dx/(x + 4 - x^2)`
`= int_0^2 dx/(4 - (x^2 - x))`
`= int_0^2 dx/(4 + 1/4 - (x - 1/2)^2)`
`= int_0^2 dx/((sqrt17/2)^2 - (x - 1/2)^2)`
`= 1/(2 xx sqrt17/2) [log (sqrt17/2 + (x - 1/2))/(sqrt17/2 - (x - 1/2)}]_0^2`
`= 1/sqrt17 [log (sqrt17 + 2x - 1)/(sqrt17 - 2x + 1)]_0^2`
`= 1/sqrt17 [log (sqrt17 + 3)/(sqrt17 - 3) - log (sqrt17 - 1)/(sqrt17 + 1)]`
`= 1/sqrt17 log [(sqrt17 + 3)/(sqrt17 - 3) xx (sqrt17 + 1)/(sqrt17 - 1)]`
`= 1/sqrt17 log [(17 +3 + 3sqrt17 + sqrt17)/(17 + 3 - 3sqrt17 - sqrt17)]`
`= 1/sqrt17 log ((20 + 4sqrt17)/(20 - 4sqrt17))`
`= 1/sqrt17 log ((5 + sqrt17)/(5 - sqrt17))`
`= 1/sqrt17 log ((5 + sqrt17)/(5 - sqrt17) xx (5 + sqrt17)/(5 + sqrt17))`
`= 1/sqrt17 log [(25 + 17 + 10sqrt17)/(25 - 17)]`
`= 1/sqrt17 log [(41 + 10 sqrt17)/8]`
`= 1/sqrt17 log [(21 + 5 sqrt17)/4]`
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