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int(sqrt(16+(logx)^(2)))/(x)dx=?...

`int(sqrt(16+(logx)^(2)))/(x)dx=?`

A

`(1)/(2) log x.sqrt(16+(logx)^(2))+8log |logx+sqrt(16+(logx)^(2))|+C`

B

`(1)/(2) log x.sqrt(16+(logx)^(2))+4log |logx+sqrt(16+(logx)^(2))|+C`

C

`(1)/(2) log x.sqrt(16+(logx)^(2))+16log |logx+sqrt(16+(logx)^(2))|+C`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
Put `log x=t and (1)/(x) dx=dt . ` then,
`I=int sqrt(t^(2)+(4)^(2))=(t)/(2)sqrt(t^(2)+16)+(16)/(2)log |t+sqrt(t^(2)+16)|+C`
`=(1)/(2)log xsqrt((log x)^(2)+16)+8log |logx+sqrt((logx)^(2)+16)|+C.`
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