`int_(0)^(2)(6x+3)/(x^(2)+4)dx`

Explanation

माना `I=int_(0)^(2)(6x+3)/(x^(2)+4)dx`
`=int_(0)^(2)(6x)/(x^(2)+4)dx+int_(0)^(2)(3)/(x^(2)+4)dx`
`{:(माना ,x^(2)+4=t,rArr.,2xdx=dt),(,x=0,rArr.,t=0+4=4),(,x=2,rArr.,t=2^(2)+4=8):}`
`therefore" "i=int_(4)^(8)(3)/(t)dt+int_(0)^(2)(3)/(x^(2)+4)dx`
`=3int_(4)^(8)(1)/(t)dt+3int_(0)^(2)(1)/(x^(2)+2^(2))dx`
`=3[logt]_(4)^(8)+(3)/(2)[tan^(-1).(x)/(2)]_(0)^(2)`
`=3[log(8)-log(4)+(3)/(2)(tan^(-1).(2)/(2))`
`=3log((8)/(4))+(3)/(2)xx(pi)/(4)`
`= 3log2+(3pi)/(8)`