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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