int(x+2)/((x^2+3x+3)sqrt(x+1))dx is equa...

`int(x+2)/((x^2+3x+3)sqrt(x+1))dx` is equal to `1/3tan^(-1)(x/(sqrt(3(x+1))))` (b) `2/(sqrt(3))tan^(-1)(x/(sqrt(3(x+1))))` `2/(sqrt(3))tan^(-1)(x/(sqrt(x+1)))` (d) none of these

`(1)/(sqrt3)"tan"^(-1)((x)/(sqrt(3(x+1))))`

`(2)/(sqrt3)"tan"^(-1)((x)/(sqrt(3(x+1))))`

`(2)/(sqrt3)"tan"^(-1)((x)/(sqrt(x+1)))`

non of these

Text Solution

Verified by Experts

The correct Answer is:

` "Let " I=int(x+2)/((x^(2)+3x+3)sqrt(x+1))dx `
` "Putting " x+1=t^(2), dx=2tdt, " we get " `
`I=2int(t^(2)+1)/(t^(4)+t^(2)+1)dt=2int(1+(1//t)^(2))/((t-(1)/(t))^(2)+3)dt `
`=(2)/(sqrt(3))"tan"^(-1)((t-(1)/(t))/(sqrt(3)))+C`
`=(2)/(sqrt(3))"tan"^(-1)((x)/(sqrt(3(x+1))))+C`

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