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#