The value of the integral `int_(0)^(2) (log(x^(2)+2))/((x+2)^(2))`, dx is

Options

A. `(sqrt(2))/(3)tan^(-1)sqrt(2)+(5)/(12)log2-(1)/(4)log3`
B. `(sqrt(2))/(3)tan^(-1)sqrt(2)-(5)/(12)log2-(1)/(12)log3`
C. `(sqrt(2))/(3)tan^(-1)sqrt(2)+(5)/(12)log2+(1)/(4)log3`
D. `(sqrt(2))/(3)tan^(-1)sqrt(2)-(5)/(12)log2+(1)/(12)log3`

Answer

D

Explanation

Let`I=overset(2)underset(0)int (log(x^(2)+2))/((x+2)^(2))`dx. Then,
`I=-[(log(x^(2)+2))/((x+2))]_(0)^(2)+overset(2)underset(0)int (2x)/((x^(2)+2)(x+2))dx`
`rArr I=-(1)/(4)log 6+(1)/(2)log2+overset(2)underset(0)int {(-2)/(3(x+2))+((2)/(3)x+(2)/(3))/(x^(2)+2)}dx`
`rArr I=(1)/(4)log2-(1)/(4)log3+[-(2)/(3)log(x+2)+(1)/(3)log(x^(2)+2)+(sqrt(2))/(3)"tan"^(-1)(x)/(sqrt(2))]_(0)^(2)`
`rArr I=(1)/(4)log2-(1)/(4)log3+{-(2)/(3)log2+(1)/(3)log3+(sqrt(2))/(3)"tan"^(-1)sqrt(2)}`
`rArr I=(-5)/(12)log2+(1)/(12)log3+(sqrt(2))/(3)"tan"^(-1)sqrt(2)`