Question

`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

Answer 1

Correct Answer - 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.`