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Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+...

`Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+A x+C ,` then `f(x)=1/2x^2` (b) `g(x)=logx` `A=1` (d) none of these

A

`f(x)=(1)/(2)x^(2)`

B

`g(x)=log x`

C

`A=1`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
D
` intx log(1+(1)/(x))dx `
`=int xlog(x+1)dx-int x logxdx `
` =(x^(2))/(2)log(x+1)-(1)/(2)int(x^(2))/(x+1)dx-(x^(2))/(2)logx+(1)/(2)int(x^(2))/(x)dx+C`
` =(x^(2))/(2)log(x+1)-(1)/(2)int(x-1+(1)/(x+1))dx-(x^(2))/(2)logx+(1)/(4)x^(2)+C`
`=(x^(2))/(2)log(x+1)-(x^(2))/(2)logx-(1)/(2)((x^(2))/(2)-x)-(1)/(2)log(x+1)+(1)/(4)x^(2)+C`
`=(x^(2))/(2)log(x+1)-(x^(2))/(2)logx-(1)/(2)log(x+1)+(1)/(2)x+C`
`"Hence, "f(x)=(x^(2))/(2)-(1)/(2),g(x)=-(1)/(2)logx, " and " A=(1)/(2).`
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