How do you evaluate the definite integral $int dx/(x(1+ln(x)))$ from $[1,e^6]$ ?

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Answer 1 · 2016-02-26T17:46:12

Is one factor the derivative of another? Then try substitution.

$d/dx(lnx) = 1/x$ , so $d/dx(1+lnx) = 1/x$ .

$int_1^(e^6) 1/(1+lnx) 1/x dx$

Let $u = 1+lnx$ , so $du = 1/x dx$ and when

$x=1$ , $u = 1$ and when

$x=e^6$ , $u = 7$

The integral becomes:

$int_1^7 1/u du = lnu]_1^7 = ln7$

How do you evaluate the definite integral int dx/(x(1+ln(x)))int dx/(x(1+ln(x))) from [1,e^6][1,e^6]?