Evaluate the definite integral. integrate from 1 to 5 (dx)/(x(1+ln(x)))

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You should use the following substitution to evaluate the given integral such that:

`lnx + 1 = t => (1/x)dx = dt`

Changing the variable yields:

`int ((dx)/x)/(1 + lnx) = int (dt)/t = ln|t| + c`

Substituting back 1 + ln x for t yields:

`int ((dx)/x)/(1 + lnx) = ln|1 + ln x| + c`

You should use the fundamental theorem of calculus to evaluate the definite integral such that:

`int_1^5 ((dx)/x)/(1 + lnx) = (ln|1 + ln x|)|_1^5`

`int_1^5 ((dx)/x)/(1 + lnx) = ln|1 + ln 5| - ln|1 + ln 1|`

You need to substitute 0 for ln 1 such that:

`int_1^5 ((dx)/x)/(1 + lnx) = ln|1 + ln 5| - ln 1`

`int_1^5 ((dx)/x)/(1 + lnx) = ln|1 + ln 5| `

`int_1^5 ((dx)/x)/(1 + lnx) = ln|ln e + ln 5|`

Using logarithmic identities yields:

`int_1^5 ((dx)/x)/(1 + lnx) = ln|ln (5e)|`

Hence, evaluating the given definite integral yields  `int_1^5 ((dx)/x)/(1 + lnx) = ln|ln (5e)|.`

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